<Li> A degenerate (or pathological) tree is where each parent node has only one associated child node . This means that performance-wise, the tree will behave like a linked list data structure . </Li> <Ul> <Li> The number of nodes n (\ displaystyle n) in a full binary tree, is at least n = 2 h + 1 (\ displaystyle n = 2h + 1) and at most n = 2 h + 1 − 1 (\ displaystyle n = 2 ^ (h + 1) - 1), where h (\ displaystyle h) is the height of the tree . A tree consisting of only a root node has a height of 0 . </Li> <Li> The number of leaf nodes l (\ displaystyle l) in a perfect binary tree, is l = (n + 1) / 2 (\ displaystyle l = (n + 1) / 2) because the number of non-leaf (a.k.a. internal) nodes n − l = ∑ k = 0 log 2 ⁡ (l) − 1 2 k = 2 log 2 ⁡ (l) − 1 = l − 1 (\ displaystyle n-l = \ sum _ (k = 0) ^ (\ log _ (2) (l) - 1) 2 ^ (k) = 2 ^ (\ log _ (2) (l)) - 1 = l - 1). </Li> <Li> This means that a perfect binary tree with l (\ displaystyle l) leaves has n = 2 l − 1 (\ displaystyle n = 2l - 1) nodes . </Li> <Li> In a balanced full binary tree, h = ⌈ log 2 ⁡ (l) ⌉ + 1 = ⌈ log 2 ⁡ ((n + 1) / 2) ⌉ + 1 = ⌈ log 2 ⁡ (n + 1) ⌉ (\ displaystyle h = \ lceil \ log _ (2) (l) \ rceil + 1 = \ lceil \ log _ (2) ((n + 1) / 2) \ rceil + 1 = \ lceil \ log _ (2) (n + 1) \ rceil) (see ceiling function). </Li> <Li> In a perfect full binary tree, l = 2 h (\ displaystyle l = 2 ^ (h)) thus n = 2 h + 1 − 1 (\ displaystyle n = 2 ^ (h + 1) - 1). </Li> <Li> The maximum possible number of null links (i.e., absent children of the nodes) in a complete binary tree of n nodes is (n + 1), where only 1 node exists in bottom-most level to the far left . </Li> <Li> The number of internal nodes in a complete binary tree of n nodes is ⌊ n / 2 ⌋ (\ displaystyle \ lfloor n / 2 \ rfloor). </Li> <Li> For any non-empty binary tree with n leaf nodes and n nodes of degree 2, n = n + 1 . </Li> </Ul> <Li> The number of nodes n (\ displaystyle n) in a full binary tree, is at least n = 2 h + 1 (\ displaystyle n = 2h + 1) and at most n = 2 h + 1 − 1 (\ displaystyle n = 2 ^ (h + 1) - 1), where h (\ displaystyle h) is the height of the tree . A tree consisting of only a root node has a height of 0 . </Li> <Li> The number of leaf nodes l (\ displaystyle l) in a perfect binary tree, is l = (n + 1) / 2 (\ displaystyle l = (n + 1) / 2) because the number of non-leaf (a.k.a. internal) nodes n − l = ∑ k = 0 log 2 ⁡ (l) − 1 2 k = 2 log 2 ⁡ (l) − 1 = l − 1 (\ displaystyle n-l = \ sum _ (k = 0) ^ (\ log _ (2) (l) - 1) 2 ^ (k) = 2 ^ (\ log _ (2) (l)) - 1 = l - 1). </Li>

Height of full binary tree with n internal nodes