Some Arithmetic, Algebraic and Combinatorial Aspects of Plane Binary Trees Raazesh Sainudiin† PartIwith:JenniferHarlow†&WarwickTucker(Maths@Uppsala-SW) PartIIwith:SeanCleary(Maths@CCNY-USA),RobertGriffiths(Stats@Oxford-UK), MareikeFischer(Maths&CompSci@Greifswald-DL)&DavidWelch(CompSci@Auckland-NZ) †SchoolofMathematicsandStatistics,UniversityofCanterbury, Christchurch,NewZealand (OnSabbatical,Dept.ofMathematics,CornellUniversity,Ithaca,NewYork,USA October 27 2014, CornellDiscreteGeometry&CombinatoricsSeminar 1/94 Part I: Arithmetic and Algebra over Plane Binary Trees Main Idea & Motivating Examples Regular 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 Trees Catalan Coefficients Split-Path Invariant Distributions Conclusions of Part II 2/94 Part I: Arithmetic and Algebra over Plane Binary Trees 3/94 2. naturally extends to arithmetic over intervals, eg. [1,2]+[3,4] = [4,6] 3. Our Main Idea: – is to further naturally extend to arithmetic over mapped partitions of an interval called Mapped Regular Pavings (MRPs) 4. – by exploiting the algebraic structure of partitions formed by rooted-plane-binary (rpb) trees 5. – thereby provide algorithms for several algebras and their inclusions over rpb tree partitions Extending Arithmetic: reals→intervals→mappedpartitionsofinterval 1. arithmetic over reals, eg. 1+3 = 4 4/94 3. Our Main Idea: – is to further naturally extend to arithmetic over mapped partitions of an interval called Mapped Regular Pavings (MRPs) 4. – by exploiting the algebraic structure of partitions formed by rooted-plane-binary (rpb) trees 5. – thereby provide algorithms for several algebras and their inclusions over rpb tree partitions Extending Arithmetic: reals→intervals→mappedpartitionsofinterval 1. arithmetic over reals, eg. 1+3 = 4 2. naturally extends to arithmetic over intervals, eg. [1,2]+[3,4] = [4,6] 5/94 4. – by exploiting the algebraic structure of partitions formed by rooted-plane-binary (rpb) trees 5. – thereby provide algorithms for several algebras and their inclusions over rpb tree partitions Extending Arithmetic: reals→intervals→mappedpartitionsofinterval 1. arithmetic over reals, eg. 1+3 = 4 2. naturally extends to arithmetic over intervals, eg. [1,2]+[3,4] = [4,6] 3. Our Main Idea: – is to further naturally extend to arithmetic over mapped partitions of an interval called Mapped Regular Pavings (MRPs) 6/94 5. – thereby provide algorithms for several algebras and their inclusions over rpb tree partitions Extending Arithmetic: reals→intervals→mappedpartitionsofinterval 1. arithmetic over reals, eg. 1+3 = 4 2. naturally extends to arithmetic over intervals, eg. [1,2]+[3,4] = [4,6] 3. Our Main Idea: – is to further naturally extend to arithmetic over mapped partitions of an interval called Mapped Regular Pavings (MRPs) 4. – by exploiting the algebraic structure of partitions formed by rooted-plane-binary (rpb) trees 7/94 Extending Arithmetic: reals→intervals→mappedpartitionsofinterval 1. arithmetic over reals, eg. 1+3 = 4 2. naturally extends to arithmetic over intervals, eg. [1,2]+[3,4] = [4,6] 3. Our Main Idea: – is to further naturally extend to arithmetic over mapped partitions of an interval called Mapped Regular Pavings (MRPs) 4. – by exploiting the algebraic structure of partitions formed by rooted-plane-binary (rpb) trees 5. – thereby provide algorithms for several algebras and their inclusions over rpb tree partitions 8/94 arithmetic from intervals to their rpb-tree partitions Figure: Arithmeticwithcolouredspaces. 9/94 arithmetic from intervals to their rpb-tree partitions Figure: Intersectionoftwohollowspheres. 10/94