Fixed layer embeddings of binary trees

We describe total congestion 1 embeddings of complete binary trees into three dimensional grids with a fixed number of layers. More specifically, we give a one-to-one embedding of any complete binary tree into a hexahedron shaped grid such that no tree nodes or edges occupy the same grid positions. With 7 layers, the number of nodes in the grid is at most 1.09375 times the number of nodes in the tree and with 5 layers we obtain a ratio of 75/64 = 1.171875. Unlike more standard embeddings these embeddings intricately weave the branches of various subtrees into each other. Finally using a standard recursive method, for 2 layers a ratio of 39/32 = 1.21875 can be obtained.