Some Arithmetic, Algebraic and Combinatorial Aspects of...

Some Arithmetic, Algebraic andCombinatorial Aspects of Plane Binary Trees

Raazesh Sainudiin†

Part I with: Jennifer Harlow† & Warwick Tucker (Maths@Uppsala-SW)

Part II with: Sean Cleary (Maths@CCNY-USA), Robert Griffiths (Stats@Oxford-UK),

Mareike Fischer (Maths&CompSci@Greifswald-DL) & David Welch (CompSci@Auckland-NZ)

†School of Mathematics and Statistics, University of Canterbury,Christchurch, New Zealand

(On Sabbatical, Dept. of Mathematics, Cornell University, Ithaca, New York, USA

October 27 2014,Cornell Discrete Geometry & Combinatorics Seminar

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Part I: Arithmetic and Algebra over Plane Binary TreesMain Idea & Motivating ExamplesRegular Pavings (RPs)Mapped Regular Pavings (MRPs)Real Mapped Regular Pavings (R-MRPs)Applications of Mapped Regular Pavings (MRPs)Conclusions of Part I

Part II: Combinatorics for Distributions over Plane Binary TreesCatalan CoefficientsSplit-Path Invariant DistributionsConclusions of Part II

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Part I: Arithmetic and Algebra over Plane Binary Trees

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Extending Arithmetic:reals→ intervals→ mapped partitions of interval

1. arithmetic over reals, eg. 1 + 3 = 4

2. naturally extends toarithmetic over intervals, eg. [1,2] + [3,4] = [4,6]

3. Our Main Idea:– is to further naturally extend toarithmetic over mapped partitions of an interval calledMapped Regular Pavings (MRPs)

4. – by exploiting the algebraic structure of partitions formedby rooted-plane-binary (rpb) trees

5. – thereby provide algorithms for several algebras and theirinclusions over rpb tree partitions

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1. arithmetic over reals, eg. 1 + 3 = 42. naturally extends to

arithmetic over intervals, eg. [1,2] + [3,4] = [4,6]

3. Our Main Idea:– is to further naturally extend toarithmetic over mapped partitions of an interval calledMapped Regular Pavings (MRPs)



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arithmetic over intervals, eg. [1,2] + [3,4] = [4,6]3. Our Main Idea:

– is to further naturally extend toarithmetic over mapped partitions of an interval calledMapped Regular Pavings (MRPs)



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arithmetic from intervals to their rpb-tree partitions

Figure: Arithmetic with coloured spaces.

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Figure: Intersection of two hollow spheres.

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Figure: Histogram averaging.

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An RP tree a root interval xρ ∈ IRd

The regularly paved boxes of xρ can be represented by nodes ofrooted-plane-binary (rpb) trees of enumerative combinatorics

finite-rooted-binary (frb) trees of geometric group theoryAn operation of bisection on a box is equivalent to performing the operation on its corresponding node in the tree:

Leaf boxes of RP tree partition the root interval xρ ∈ IR2

zρ

xρ

zρzρL

zρR

��

@@@

xρL xρR

zρ�

��z

��zρLL

@@@zρLR

@@@zρR

xρLR

xρLL

xρR

zρ��z

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@@@zρRR

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xρLR

xρLL xρRL

xρRR

zρ�

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@@@z

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AAA

z��

AAAz

ρLL

z zρRL

zρRR

��

AAAz

ρLRL

zρLRR

xρLR

L

xρLR

R

xρLL xρRL

xρRR

By this “RP Peano’s curve” rpb-trees encode paritions of xρ ∈ IRd

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~ρ

xρ

~ρ

~ρL

~ρR

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@@@@@@

xρL xρR

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@@@@@@~ρR

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zρ

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ρLRL

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zρ

xρ

zρzρL

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@@@

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zρ

xρ

zρzρL

zρR

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@@@

xρL xρR

zρ�

��z

��zρLL

@@@zρLR

@@@zρR

xρLR

xρLL

xρR

zρ��z

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xρLR

xρLL xρRL

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AAA

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z zρRL

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ρLRL

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xρLR

L

xρLR

R

xρLL xρRL

xρRR

By this “RP Peano’s curve” rpb-trees encode paritions of xρ ∈ IRd15 / 94

Algebraic Structure and Combinatorics of RPs

Leaf-depth encoded RPs

There are Ck RPs with k splits

C0 = 1C1 = 1C2 = 2C3 = 5C4 = 14C5 = 42. . . = . . .

Ck =(2k)!

(k+1)!k!. . . = . . .C15 = 9694845. . . = . . .C20 = 6564120420. . . = . . .

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Hasse (transition) Diagram of Regular Pavings

Transition diagram over S0:3 with split/reunion operations

RS, W.Taylor and G.Teng, Catalan Coefficients, Sequence A185155 in The On-Line Encyclopedia of Integer

Sequences, 2012, http://oeis.org

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http://oeis.org

Hasse (transition) Diagram of Regular Pavings

Transition diagram over S0:4 with split/reunion operations

1. The above state space is denoted by S0:42. Number of RPs with k splits is the Catalan number Ck3. There is more than one way to reach a RP by k splits4. Randomized enclosure algorithms are Markov chains on

S0:∞

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RPs are closed under union operations

s(1) ∪ s(2) = s is union of two RPs s(1) and s(2) of xρ ∈ R2.

s(1)

zρ(1)�

��

@@@zρ(1)L

��

@@@zρ(1)LL zρ(1)LR

zρ(1)R

s(2)

zρ(2)�

��

@@@zρ(2)L zρ(2)R�

��

@@@zρ(2)RL zρ(2)RR

s

zρ�

��

@@@zρL

��

@@@zρLL zρLR

zρR�

��@@@

zρRL

zρRR

xρ(1)LR

xρ(1)LL

xρ(1)R

∪

xρ(2)RR

xρ(2)RL

xρ(2)L

=

xρLR

xρLL xρRL

xρRR

Lemma 1: The algebraic structure of rpb-trees (underlyingThompson’s group) is closed under union operations.

Proof: by a “transparency overlay process” argument (cf. Meier2008).


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Algorithm 1: RPUnion(ρ(1), ρ(2))input : Root nodes ρ(1) and ρ(2) of RPs s(1) and s(2) , respectively, with root box x

ρ(1)= x

ρ(2)

output : Root node ρ of RP s = s(1) ∪ s(2)

if IsLeaf(ρ(1)) & IsLeaf(ρ(2)) thenρ← Copy(ρ(1))return ρ

end

else if !IsLeaf(ρ(1)) & IsLeaf(ρ(2)) thenρ← Copy(ρ(1))return ρ

end

else if IsLeaf(ρ(1)) & !IsLeaf(ρ(2)) thenρ← Copy(ρ(2))return ρ

end

else!IsLeaf(ρ(1)) & !IsLeaf(ρ(2))

endMake ρ as a node with xρ ← xρ(1)Graft onto ρ as left child the node RPUnion(ρ(1)L, ρ(2)L)Graft onto ρ as right child the node RPUnion(ρ(1)R, ρ(2)R)return ρ

Note: this is not the minimal union of the (Boolean mapped) RPs of Jaulin et. al. 2001

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Dfn: Mapped Regular Paving (MRP)

I Let s ∈ S0:∞ be an RP with root node ρ and root boxxρ ∈ IRd

I and let Y be a non-empty set.I Let V(s) and L(s) denote the sets all nodes and leaf nodes

of s, respectively.I Let f : V(s)→ Y map each node of s to an element in Y as

follows:{ρv 7→ fρv : ρv ∈ V(s), fρv ∈ Y} .

I Such a map f is called a Y-mapped regular paving(Y-MRP).

I Thus, a Y-MRP f is obtained by augmenting each node ρvof the RP tree s with an additional data member fρv.

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I and let Y be a non-empty set.

I Let V(s) and L(s) denote the sets all nodes and leaf nodesof s, respectively.

I Let f : V(s)→ Y map each node of s to an element in Y asfollows:

{ρv 7→ fρv : ρv ∈ V(s), fρv ∈ Y} .



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of s, respectively.

I Let f : V(s)→ Y map each node of s to an element in Y asfollows:

{ρv 7→ fρv : ρv ∈ V(s), fρv ∈ Y} .



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Examples of Y-MRPs

If Y = R

R-MRP over s221 with xρ = [0,8]

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Examples of Y-MRPs

If Y = B

B-MRP over s122 with xρ = [0,1]2 (e.g. Jaulin et. al. 2001)

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Examples of Y-MRPs

If Y = IR– rpb tree representation for interval inclusion algebra

IR-MRP enclosure of the Rosenbrock function withxρ = [−1,1]2

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Examples of Y-MRPs

If Y = [0,1]3– R G B colour maps

[0,1]3-MRP over s3321 with xρ = [0,1]3

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Examples of Y-MRPs

If Y = Z+ := {0,1,2, ...}– radar-measured aircraft trajectory data

Z+-MRP trajectory of an aircraft and its tree

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Examples of Y-MRPs

If Y = S2, xρ = [0,1]2 – vector-MRPs

Two Views of f

Two Views of g

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Y-MRP Arithmetic

If ? : Y× Y→ Y then we can extend ? point-wise to twoY-MRPs f and g with root nodes ρ(1) and ρ(2) viaMRPOperate(ρ(1), ρ(2), ?).This is done using MRPOperate(ρ(1), ρ(2),+)

f g f + g

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R-MRP Addition by MRPOperate(ρ(1), ρ(2),+)

adding two piece-wise constant functions or R-MRPs

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Algorithm 2: MRPOperate(ρ(1), ρ(2), ?)input : two root nodes ρ(1) and ρ(2) with same root box x

ρ(1)= x

ρ(2)and binary operation ?.

output : the root node ρ of Y-MRP h = f ? g.Make a new node ρ with box and imagexρ ← xρ(1) ; hρ ← fρ(1) ? gρ(2)

if IsLeaf(ρ(1)) & !IsLeaf(ρ(2)) thenMake temporary nodes L′, R′

xL′ ← xρ(1)L; xR′ ← xρ(1)RfL′ ← fρ(1) , fR′ ← fρ(1)Graft onto ρ as left child the node MRPOperate(L′, ρ(2)L, ?)Graft onto ρ as right child the node MRPOperate(R′, ρ(2)R, ?)

end

else if !IsLeaf(ρ(1)) & IsLeaf(ρ(2)) thenMake temporary nodes L′, R′

xL′ ← xρ(2)L; xR′ ← xρ(2)RgL′ ← gρ(2) , gR′ ← gρ(2)Graft onto ρ as left child the node MRPOperate(ρ(1)L, L′, ?)Graft onto ρ as right child the node MRPOperate(ρ(1)R,R′, ?)

end

else if !IsLeaf(ρ(1)) & !IsLeaf(ρ(2)) thenGraft onto ρ as left child the node MRPOperate(ρ(1)L, ρ(2)L, ?)Graft onto ρ as right child the node MRPOperate(ρ(1)R, ρ(2)R, ?)

endreturn ρ

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Unary transformations are easy too

Let MRPTransform(ρ, τ) apply the unary transformationτ : R→ R to a given R-MRP f with root node ρ as follows:

I copy f to gI recursively set fρv = τ(fρv ) for each node ρv in gI return g as τ(f )

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Minimal Representation of R-MRP

Algorithm 3: MinimiseLeaves(ρ)input : ρ, the root node of R-MRP f .output : Modify f into h(f ), the unique R-MRP with fewest leaves.if !IsLeaf(ρ) then

MinimiseLeaves(ρL)MinimiseLeaves(ρR)

if IsCherry(ρ) & ( fρL = fρR ) thenfρ ← fρLPrune(ρL)Prune(ρR)

endend

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Arithmetic and Algebra of R-MRPs

Thus, we can obtain arithmetical expressions specified byfinitely many sub-expressions in a directed acyclic graphwhose:

I inputs and output nodes are themselves R-MRPsI and whose edges involve:

1. a binary arithmetic operation ? ∈ {+,−, ·, /} over twoR-MRPs,

2. a standard transformation of R-MRP by elements ofS := {exp, sin, cos, tan, . . .} and

3. their compositions.

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I inputs and output nodes are themselves R-MRPs

I and whose edges involve:1. a binary arithmetic operation ? ∈ {+,−, ·, /} over two

R-MRPs,2. a standard transformation of R-MRP by elements of

S := {exp, sin, cos, tan, . . .} and


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Stone-Weierstrass Theorem: R-MRPs Dense in C(xρ,R)

TheoremLet F be the class of R-MRPs with the same root box xρ. ThenF is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.

Proof:Since xρ ∈ IRd is a compact Hausdorff space, by theStone-Weierstrass theorem we can establish that F is dense inC(xρ,R) with the topology of uniform convergence, providedthat F is a sub-algebra of C(xρ,R) that separates points in xρand which contains a non-zero constant function.

We will show all these conditions are satisfied by F

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TheoremLet F be the class of R-MRPs with the same root box xρ. ThenF is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.Proof:Since xρ ∈ IRd is a compact Hausdorff space, by theStone-Weierstrass theorem we can establish that F is dense inC(xρ,R) with the topology of uniform convergence, providedthat F is a sub-algebra of C(xρ,R) that separates points in xρand which contains a non-zero constant function.


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TheoremLet F be the class of R-MRPs with the same root box xρ. ThenF is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.Proof:Since xρ ∈ IRd is a compact Hausdorff space, by theStone-Weierstrass theorem we can establish that F is dense inC(xρ,R) with the topology of uniform convergence, providedthat F is a sub-algebra of C(xρ,R) that separates points in xρand which contains a non-zero constant function.


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Stone-Weierstrass Theorem Contd.: R-MRPs Dense inC(xρ,R)

I F is a sub-algebra of C(xρ,R) since it is closed underaddition and scalar multiplication.

I F contains non-zero constant functionsI Finally, RPs can clearly separate distinct points x , x ′ ∈ xρ

into distinct leaf boxes by splitting deeply enough.I Thus, F , the class of R-MRPs with the same root box xρ,

is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.

I Q.E.D.

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I F contains non-zero constant functions

I Finally, RPs can clearly separate distinct points x , x ′ ∈ xρinto distinct leaf boxes by splitting deeply enough.

I Thus, F , the class of R-MRPs with the same root box xρ,is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.

I Q.E.D.

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into distinct leaf boxes by splitting deeply enough.

I Thus, F , the class of R-MRPs with the same root box xρ,is dense in C(xρ,R), the algebra of real-valued continuousfunctions on xρ.

I Q.E.D.

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I Q.E.D.

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I Q.E.D.

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Approximating Kernel Density Estimates by R-MRPs

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Nonparametric Density Estimation

Problem: Take samples from an unknown density f and consistentlyreconstruct f

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Approach: Use statistical regular paving to get R-MRP data-adaptivehistogram

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Solution: R-MRP histogram averaging allows us to produce aconsistent Bayesian estimate of the density (up to 10 dimensions)(Teng, Harlow, Lee and S., ACM Trans. Mod. & Comp. Sim., [r. 2] 2012)

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Coverage, Marginal & Slice Operators of R-MRP

R-MRP approximation to Levy density and its coverage regions withα = 0.9 (light gray), α = 0.5 (dark gray) and α = 0.1 (black)

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Marginal densities f {1}(x1) and f {2}(x2) along each coordinate ofR-MRP approximation

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The slices of a simple R-MRP in 2D

— “non-parametric regression arithmetic”

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[0,1]3-MRP Arithmetic over Colored Cubes

f g

f + g f − g

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Slices of colored cube

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B-MRP arithmetic – contractors, propagators &collaborators (Comp-aided Proofs in Anal./Dyn.)

Two Boolean-mapped regular pavings A1 and A2 and Booleanarithmetic operations with + for set union, − for symmetric set

difference, × for set intersection, and ÷ for set difference.

A1 A2 A1 + A2

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B-MRP arithmetic – contractors, propagators &collaborators (Comp-aided Proofs in Anal./Dyn.)

Two Boolean-mapped regular pavings A1 and A2 and Booleanarithmetic operations with + for set union, − for symmetric set

difference, × for set intersection, and ÷ for set difference.

A1 − A2 A1 × A2 A1 ÷ A2

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Vector-MRP arithmetic

If Y = S2, xρ = [0,1]2 – vector-MRPs

Two Views of f

Two Views of g

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Vector-MRP arithmetic

f × g — cross-product of vector-MRPs

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Air Traffic “Arithmetic”

(G. Teng, K. Kuhn and RS, J. Aerospace Comput., Inf. & Com., 9:1, 14–25, 2012.)

On a Good Day

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Z+-MRP On a Good Day

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On a Bad Day

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Z+-MRP On a Bad Day

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Z-MRP pattern for Good Day − Bad Day

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MRS 1.0: A C++ Class Library for Statistical Set Processing,

Harlow, S & York, 2013

MRS 1.0 is GNU auto-confiscated, Doxygenized, GPL-licensed (builds on GNU Sci. Lib., C-XSC & Boost) and has:

Its is templatised and can be extended to general Y-MRPs.

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Conclusions of Part I

I Y-MRPs provide rpb-tree partition arithmeticI IY-MRPs allow efficient arithmetic for Neumaier’s inclusion

algebrasI I IY can be IR for f : IRd → IR

I IY can be IRm for f : IRd → IRmI IY can be (IR, IRm, IRm2) for range, gradient & Hessian of

f : IRd → IRI Other obvious extensions include arithmetic over Taylor

polynomial inclusion algebrasI In general the domain and range of f can be complete

lattices with intervals and bisection operationsI We have seen several statistical applications of Y-MRPsI CODE: mrs: a C++ class library for statistical set

processing by Harlow, S and York.

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Part II: Combinatorics for Distributions over Plane Binary Trees

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Catalan Coefficients

How many distinct “splitting paths” are there from the root nodeto a given rpb-tree T ?Let this be B(T ), the Catalan Coefficient of T .

1 1 2 1 1

There are Ck RPs with k splits

and k! distinct paths to them

k Ck k!0 1 11 1 12 2 23 5 64 14 245 42 120. . . . . . . . .

k (2k)!(k+1)!k! k!

. . . . . . . . .

1 1 2 1 1

For example: B((2,2,2,2)) = 2 and B((3,3,2,1)) = 1

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Catalan coefficients – OEIS A185155

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Catalan coefficients of rpb-trees with 3,4,5 splits

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Frequency of Catalan coefficients of rpb-trees with6,7,8 splits

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Catalan coefficient of a rpb-tree

Let an interior node of T include the root node and exclude allleaf nodes of T . Then the Catalan coefficient of T is:

B(T ) =|T |!∏

v∈T|Tv |

=(# of interior nodes of T )!∏

v∈T# of interior nodes of sub-tree of T with root v

Proof:

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B(T ) =|T |!∏

v∈T|Tv |



Proof:Let L(Tv ) and R(Tv ) be left and right sub-trees of Tv with rootnode v . Then the number of distinct binary inter-leavingsbetween the interior (split) nodes of L(Tv ) and R(Tv ) is:(

|L(Tv )|+ |R(Tv )||L(Tv )|

)=

(|L(Tv )|+ |R(Tv )|)!|L(Tv )|!× |R(Tv )|!

=|Tv | × (|L(Tv )|+ |R(Tv )|)!|Tv | × |L(Tv )|!× |R(Tv )|!

=|Tv |!

|Tv | × |L(Tv )|!× |R(Tv )|!

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B(T ) =|T |!∏

v∈T|Tv |



Proof:And the number of distinct binary inter-leavings between theinterior (split) nodes of L(Tv ) and R(Tv ) as well as theirsub-trees and their sub-sub-trees and so on is:

|Tv |!|Tv | ×��|L(Tv )|!×��

��|R(Tv )|!×

��|L(Tv )|!

|L(Tv )| × |L(L(Tv ))|!× |R(L(Tv ))|!×

��|R(Tv )|!

|R(Tv )| × |L(R(Tv ))|!× |R(R(Tv ))|!× · · · 1!

1× 0!× 0!

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B(T ) =|T |!∏

v∈T|Tv |



Proof:And the number of distinct binary inter-leavings between theinterior (split) nodes of L(Tv ) and R(Tv ) as well as theirsub-trees and their sub-sub-trees and so on is:

|Tv |!|Tv | ×��|L(Tv )|!×��

��|R(Tv )|!×

��|L(Tv )|!

|L(Tv )| ×��|L(L(Tv ))|!×��

��|R(L(Tv ))|!×

��|R(Tv )|!

|R(Tv )| ×��|L(R(Tv ))|!×(((((

(|R(R(Tv ))|!× · · · ��1!

1×��0!×��0!

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B(T ) =|T |!∏

v∈T|Tv |



Proof:

B(Tv ) =|Tv |!

|Tv | × |L(Tv )|!× |R(Tv )|!× B(L(Tv ))× B(R(Tv ))

=|Tv |!

|Tv | × |L(Tv )| × |R(Tv )| × |L(L(Tv ))| × |R(L(Tv ))| × · · ·

=|Tv |!∏

u∈Tv|Tu|

Therefore,

B(T ) =|T |!∏

v∈T |Tv |83 / 94

An Example Catalan coefficient calculation

Consider the perfectly balanced tree (3,3,3,3,3,3,3,3) withk = 7 splits and 8 leaves (all with depth 3).

Then B((3,3,3,3,3,3,3,3))

=7!

7× 3× 3× 1× 1× 1× 1=

2��6× 5× 4×��3× 2

��3×��3= 80.

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Planar Yule Model

Model (Planar Yule)A rpb-tree with k splits is obtained from one with k − 1 splits bysplitting one of the k leaves uniformly at random.

Pr(T ) = B(T )1k !

This model also gives:I the probability of a binary search tree with random input

given by the unifrom distribution on all permutations of [k ](Fill, 1994)

I the probability of an planar Yule tree (non-planar case isthe Yule speciation model in Phylogenetics)

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Planar Yule Model

Model (Planar Yule)A rpb-tree with k splits is obtained from one with k − 1 splits bysplitting one of the k leaves uniformly at random.

Pr(T ) = B(T )1k !

This model also gives:I the probability of a binary search tree with random input

given by the unifrom distribution on all permutations of [k ](Fill, 1994)

I the probability of an planar Yule tree (non-planar case isthe Yule speciation model in Phylogenetics)

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f -Splitting Model

Model (f -Splitting)Let f be a probability density function on [0,1]. We canrepresent the rpb-tree by the corresponding dyadic partition of[0,1]. We split a node v of an rpb-tree corresponding to theinterval [v , v ] = [2−i ,2−j ] with probability

∫ vv f (x)dx.

Pr(T ) = B(T )∏v∈T

(∫ vv

f (x)dx

)

This model also gives:I the Statistically equivalent block rule in density estimationI gives a flexible way to explore the tree shapes,

eg. f ∼ Beta(α, β)

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f -Splitting Model

Model (f -Splitting)Let f be a probability density function on [0,1]. We canrepresent the rpb-tree by the corresponding dyadic partition of[0,1]. We split a node v of an rpb-tree corresponding to theinterval [v , v ] = [2−i ,2−j ] with probability

∫ vv f (x)dx.

Pr(T ) = B(T )∏v∈T

(∫ vv

f (x)dx

)

This model also gives:I the Statistically equivalent block rule in density estimationI gives a flexible way to explore the tree shapes,

eg. f ∼ Beta(α, β)

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Split-Path Invariant Distributions

Model (Split-Path Invariant)

Pr(T ) = B(T )× Pr{any split-path to T from root node}

The Planar Yule Model and f -Splitting Model are examples ofSplit-Path Invariant Distributions. Such distributions onrpb-trees can also be used to model:

I infection trees in epidemics on various hidden contactgraphs (eq. all-to-all, one-super-hub, etc.)

I causal trees underlying self-exciting point processesI by counting planar tree paths that map to non-palanar tree

paths we can induce these distributions on non-planartrees

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Split-Path Invariant Distributions

Model (Split-Path Invariant)

Pr(T ) = B(T )× Pr{any split-path to T from root node}

The Planar Yule Model and f -Splitting Model are examples ofSplit-Path Invariant Distributions. Such distributions onrpb-trees can also be used to model:

I infection trees in epidemics on various hidden contactgraphs (eq. all-to-all, one-super-hub, etc.)

I causal trees underlying self-exciting point processesI by counting planar tree paths that map to non-palanar tree

paths we can induce these distributions on non-planartrees

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Conclusions of Part II

I Several probablistic models on rooted planar binary treesexist

I Using the notion of split-path invariance and Catalancoefficients we can specify various useful distributions onrpb-trees

I These distributions can also be further projected ontonon-planar binary trees

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References

Jaulin, L., Kieffer, M., Didrit, O. & Walter, E., Applied intervalanalysis. London: Springer-Verlag, 2001.Meier, J., Groups, graphs and trees: an introduction to thegeometry of infinite groups, CUP, Cambridge, 2008.Neumaier, A., Interval methods for systems of equations, CUP,Cambridge, 1990.Harlow, J., Sainudiin, R. & Tucker, W., Mapped RegularPavings, Reliable Computing, vol. 16, pp. 252-282, 2012.Fill, J.A., On the stationary distribution for the move-to-rootMarkov chain for binary search trees, Tech. Rep. 537,Dept. Math.Sci., Johns Hopkins Univ., 1994.Sainudiin, R., Taylor, W., & Teng, G., Catalan Coefficients,Sequence A185155 in The On-Line Encyclopedia of IntegerSequences, published electronically at http://oeis.org,2012

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http://oeis.org

Acknowledgements

I RS’s external consulting revenues from the New ZealandMinistry of Tourism

I WT’s Swedish Research Council Grant 2008-7510 thatenabled RS’s visits to Uppsala in 2006 and 2009

I Erskine grant from University of Canterbury that enabledWT’s visit to Christchurch in 2011 & 2014

I EU Marie Curie International Research Staff ExchangeGrants (CORCON 2014-2017 & CONSTRUMATH2009-2012)

I University of Canterbury MSc Scholarship to JH.

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Thank you!

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Part I: Arithmetic and Algebra over Plane Binary TreesMain Idea & Motivating ExamplesRegular Pavings (RPs)Mapped Regular Pavings (MRPs)Real Mapped Regular Pavings (R-MRPs)Applications of Mapped Regular Pavings (MRPs)Conclusions of Part I

Part II: Combinatorics for Distributions over Plane Binary TreesCatalan CoefficientsSplit-Path Invariant DistributionsConclusions of Part II

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