Full Trees

One can prove the following propositions: (1) For every set D and for every finite sequence p and for every natural number n such that p ∈ D holds p↾ Segn ∈ D. (2) For every binary tree T holds every element of T is a finite sequence of elements of Boolean. Let T be a binary tree. We see that the element of T is a finite sequence of elements of Boolean. Next we state several propositions: (3) For every tree T such that T = {0, 1} holds T is binary. (4) For every tree T such that T = {0, 1} holds Leaves(T ) = ∅. (5) Let T be a binary tree, n be a natural number, and t be an element of T . If t ∈ T -level(n), then t is a tuple of n and Boolean. (6) For every tree T such that for every element t of T holds succ t = {t a 〈0〉, t a 〈1〉} holds Leaves(T ) = ∅. (7) For every tree T such that for every element t of T holds succ t = {t a 〈0〉, t a 〈1〉} holds T is binary. (8) For every tree T holds T = {0, 1} iff for every element t of T holds succ t = {t a 〈0〉, t a 〈1〉}.