A way to convert a binary tree into a grid Wondering if there is a way to convert a binary tree into a grid such as this:

                    a
                  /   \
                 b     c
                / \   / \
               d   e f   g


                    or


                 d--b--e
                    |
                    a
                    |
                 f--c--g


                    to


                 d--b--e
                 |  |  |
                 y--a--x
                 |  |  |
                 f--c--g


An arbitrary sized binary tree. I'm not super familiar with binary tree properties so I am not sure if this would even be possible or if edge cases would prevent it from being possible.
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).