Embedding different graphs, especially binary trees, in the hypercube has a huge literature. However, I could not find anything if we restrict the embedding to be monotone. So I would like to injectively map the vertices of a complete binary tree, $T_d$, which I denote by the binary sequences of length at most $d$, into the hypercube, $C_n$, whose vertices I denote by the binary sequences of length $n$, such that if $x$ and $xb$ (where $b$ is a bit) are two adjacent vertices of $T_d$, then for their images, $f(x)$ and $f(xb)$, it holds that $f(xb)$ has more 1's than $f(x)$ and it has a $1$ everywhere where $f(x)$ has a one. So e.g., $01$ and $011$ might be mapped to $100$ and $101$ or even to $100$ and $111$. (So I do not need that they are adjacent in $C_n$.)
Our goal is to find the smallest $n$ for which such an embedding is possible.
I think I can prove some upper and lower bounds (like $n\approx 1.29d$) but I wonder if this problem has been studied before. It seems like a natural generalization of a very well studied problem (at least if we suppose that the images are adjacent in $C_n$, which should not make a difference in the asymptotics). Anyone every heard of this problem?