For a rooted tree T of order n, what is the probability of that T contains a balanced tree of order 7? For example, there are total of 719 rooted trees of order 10, IF among all those 719 rooted trees, there are 500 rooted trees which contain a subtree of balanced tree of 7, then probability is 500/719 for order n=10. What about n=100, 1000, ... As n approaches to infinity, can we assert that the probability approaches to 1?
I submitted a paper to a journal, one referee claimed the above statement to be true without proof, by which he/she said my results are not new. I could not find any paper to support his/her statement. It is a research problem, and I wish someone here can help me.