Good Subtrees

$T$ is a binary tree with sufficiently many nodes. Assume that sufficiently many of them have been colored red.

Call a subtree of $T$ good if it has at least $n$ nodes.

Choose the option with the tightest relation between $n$ and $x$ such that the following statement holds:

Given that a good subtree of $T$ exists, there always exists a good subtree with $< x$ red nodes

Bonus: Give an algorithm to find such a subtree.