The first part of this talk will exploit this property to implement arithmetic operations recursively over such trees. Depending on interest, I'll give some insights into the algebra and arithmetic of plane binary trees for purposes of (1) statistical operations in non-parametric regression and density estimation (2) rigorous operations over inclusion algebras in computer-aided proofs in analysis. The second part will describe a family of 'split-exchangeable' distributions on these trees using 'Catalan coefficients', which give the number of distinct ways in which you can obtain any plane binary tree by sequentially splitting the leaves from the root node. This is analogous to how the binomial distribution is obtained from the binomial coefficients. I will try to motivate why these distributions on tree spaces arise naturally in (i) some consistent density estimation rules, (ii) biodiversity models of speciation and extinction in macro-evolution, (iii) infection trees in epidemics on various hidden contact graphs and (iv) causal trees underlying self-exciting point processes, among others.