# Why not use large $k$ in a $k$-ary tree?

Obviously binary trees are great because of $O(\log_2 n)$ search, inserts, and deletes in best case.

To "maximize" occurrence of best case, we can use self-balancing trees like red-black trees, AVLs, splay trees, etc.

If we use a $k$-ary tree, we can get $O(\log_k n)$ searches, inserts, and deletes in best case. Similarly we can "maximize" the occurrences by using self-balancing trees.

I know these are asymptotically of the same order, i.e. $O(\log n)$, but the constant could be huge in practice.

The huge reason I can think of is that by using larger $k$'s, you lose the ability to make a decision based on one comparison.

What are other reasons larger $k$ $k$-ary trees aren't used in practice?

BTrees are used in practice - file systems, database with $k$ for example equal 1024 or 4096, so it seems to be bigger than binary.
For example ternary tree with comparing >, =, < gives theoretical improve $O(log_3 n)$, but self balancing trees in basic form do not accept duplicates, so there is more overhead and not strictly practical.