We consider finite rooted ordered trees in which every internal node has at least two children, sometimes called Schröder trees; the size |t| of such a tree t is the number of its leaves. An important concept with trees is that of inducing subtrees. Given a tree t of size k and a larger tree t’ of size n\geq k we define 0 \leq \theta(t,t’)\leq 1 to be the probability of obtaining t as a randomly induced subtree of size k in t’. One can think of \theta(t,t’) to be the density of the pattern t in t’. In this talk we consider two closely related questions concerning the nature of \theta:
1. A sequences of trees (t_n)_n with |t_n|\rightarrow\infty is called \theta-convergent, if \theta(t,t_n) converges for every fixed tree t. The limit of (t_n)_n is the function t\mapsto \lim_n\theta(t,t_n). What limits exist?
2. A Markov chain (X_n)_n with X_n being a random tree of size n is called a \theta-chain if P(X_k=t|X_n=t’)=\theta(t,t’) for all k \leq n. What \theta-chains exist?
Similar questions have been treated for many different types of discrete structures (words, permutations, graphs \dots); binary Schröder trees (Catalan trees) are considered in . We present a De Finetti-type representation for \theta-chains and a homeomorphic description of limits of \theta-convergent sequences involving certain tree-like compact subsets of the square [0,1]^2. Questions and results are closely linked to the study of exchangeable hierarchies, see .
 Evans, Grübel and Wakolbinger. “Doob-Martin boundary of Rémy’s tree growth chain”. The Annals of Probability, 2017.
 Forman, Haulk and Pitman. “A representation of exchangeable hierarchies by sampling from random real trees”. Prob.Theory and related Fields, 2017.
 Gerstenberg. “Exchangeable interval hypergraphs and limits of ordered discrete structures”. arXiv, 2018.