Subtrees 1

The concepts of root tree, the set of successors of a node in decorated tree and sets of subtrees are introduced. Let us mention that every tree which is finite is also finite-order. Next we state three propositions: (1) For every decorated tree t holds tε N = t. (2) For every tree t and for all finite sequences p, q of elements of N such that p q ∈ t holds t(p q) = tpq. (3) Let t be a decorated tree and p, q be finite sequences of elements of N. If p q ∈ domt, then t(p q) = tpq. Let I 1 be a decorated tree. We say that I 1 is root if and only if: (Def. 1) dom I 1 = the elementary tree of 0. One can check that every decorated tree which is root is also finite. We now state three propositions: (4) For every decorated tree t holds t is root iff / 0 ∈ Leaves(domt). (5) For every tree t and for every element p of t holds tp = the elementary tree of 0 iff p ∈ Leaves(t). (6) For every decorated tree t and for every node p of t holds tp is root iff p ∈ Leaves(domt). Let us observe that there exists a decorated tree which is root and there exists a decorated tree which is finite and non root. Let x be a set. One can verify that the root tree of x is finite and root. Let I 1 be a tree. We say that I 1 is finite-branching if and only if: 1 This article has been worked out during the visit of the author in Nagano in Summer 1994.