# Is it always possible to create a minimum depth tree where tree nodes are unique and have a 'choice'?

This is a somewhat long question thus sincere apologies beforehand. In a tree a node is a simple point. Now instead of a node let us consider a set of 'choice nodes' that have the following properties:

1. Each 'node' has a top side, bottom side, and a unique set of K distinct color pairs (vertical top-bottom pairs). Each color always pairs with exactly one other color (eg: red with pink, orange with yellow and so on).
2. Each 'node' can connect to another via an edge only b/w two same colors: edge b/w bottom of upper node and top of lower node (in a tree).
3. In any 'node' both colors of a pair cannot take part in connections. These connections are illegal.

Given a set of $$C$$ color pairs, each color pair occurring exactly $$2k$$ times in total: $$k$$ times as $$C_{red}:C_{pink}$$ and $$k$$ times as $$C_{pink}:C_{red}$$. Each node has $$p$$ color pairs (4 in example). Thus total number of Nodes given to us are $$N = 2kC/p$$

Query 1: Given any set of such nodes, can we always create a tree $$T$$ such that: $$Depth(T) \leq log_{p-1}N$$? In other words, can we always create a tree with smallest depth possible? I believe the answer is yes, but I am struggling to formally prove it. If NO can someone explain why (with an explicit example if possible).

Query 2: Assuming answer is YES, what algo can be used to create this minimum depth tree $$T$$?