Let $T$ be a complete infinite rooted binary tree. Is it possible to remove (infinitely many) subtrees of $T$ and get a subgraph $G$ such that:

1. $G$ has no complete subtrees (the graph below any vertex of $G$ is not a complete binary tree).

2. There exists some $\epsilon > 0$ such that for any $n \in \mathbb{N}$ the number of vertices of $G$ whose distance from the root is $n$ is at least $\epsilon2^n$.