(Note: I couldn't put balanced tree in the tags ; this topic isn't about balanced search trees, just balanced trees.)Let $A$ a binary tree.We have the relation: $log(|A|+1) \leq h(A) \leq |A|-1$ where $|A|$ denotes the size of A (the number of nodes) and $h(A)$ its height.Then I was told that A is balanced if $h(A) = O(log|A|)$. I don't really understand what that means. I know the symbol $O$ when we use it in maths with $n$ stuff. For exemple $u_n = O(v_n)$ when $n \to +\infty$ or $f = O(g)$ at a certain point $x$. But there, how can we told that the tree $A$ evolves? Applying the definition, it would mean that $\frac{|A|}{log|A|} \leq M$ but for a given tree, this is always true. So do we fix set the tree's size and have the relation for any size or do we fix the size and have the relation for any height, or something else?Thanks.
About the definition of a balanced tree
binary trees;balanced search trees