MML Identifier: TREES 9.

One can check that every tree which is finite is also finite-order. The following propositions are true: (1) For every decorated tree t holds t ε = t. (2) For every tree t and for all finite sequences p, q of elements of such that p q ∈ t holds t (p q) = t p q. (3) Let t be a decorated tree and let p, q be finite sequences of elements of . If p q ∈ dom t, then t (p q) = t p q. A decorated tree is root if: (Def.1) dom it = the elementary tree of 0. Let us note that every decorated tree which is root is also finite. The following three propositions are true: (4) For every decorated tree t holds t is root iff ε ∈ Leaves(dom t). (5) For every tree t and for every element p of t holds t p = the elementary tree of 0 iff p ∈ Leaves(t).