# How proof of Hoffman algorithm greedy property starts with optimal tree T?

In this paper Claim 1 states that x and y are smallest probability and there is optimal code tree in which this two characters are siblings at the maximum depth. In proof to that claim, author starts with tree T which has b and c at the maximum depth of tree. I can't understand why tree T is called optimal, because if x and y have smallest probabilities, then tree T cannot be optimal because Hoffman algorithm won't build such tree. I think my problem here is with understanding what optimal tree means.

If someone comes from CSLR this is about lemma 16.2 and proof to that.

Here is a formal definition of an optimal code tree. Let $$\pi$$ be a distribution. The $$\pi$$-cost of a code tree is the expected length of a codeword, where the expectation is taken with respect to $$\pi$$. A code tree is optimal for $$\pi$$ if its $$\pi$$-cost is equal to the minimum $$\pi$$-cost of any code tree.