A way to convert a binary tree into a grid
A vertex in the grid has $O(d)$ vertices at distance $d$ from it. In contrast, the complete binary tree of height $h$ as $2^h$ vertices at distance $h$ from the root. Therefore there is no isometric embedding of arbitrary binary trees in the grid. Similar considerations show that there is also no embedding of bounded distortion (in which distances increase by at most a constant factor).