Coloring a tree with a minimal sum of colors

I think three colors are enough for this. How to prove it?

This is not true. We describe the root tree algebraically as follows. V- a tree with one node tree. E(t1, t2)is a tree consisting of t1and t2and an edge from root t1to root t2, root to root t2. The next tree t3requires four colors to reach a minimum of 156.

t3 = E(t2, E(t2, E(t2, E(t2, t2))))
t2 = E(t1, E(t1, E(t1, E(t1, t1))))
t1 = E(t0, E(t0, E(t0, E(t0, t0))))
t0 = V

, , , , , .

d & ge; k & ge; 3, T (d, k) k . T (d, 1) . i > 1 T (d, i) - d , T (d, - 1).

k. k = 3 - , , , 3 . k > 3 T (d, k), k - 1 . , k . - 1, , k d > k - 1 1. 1, , 1, 1. , 1, > 1. , T (d, k - 1), .

data Tree = V | E Tree Tree
    deriving (Eq, Show)

otherMinimums [x, y] = [y, x]
otherMinimums (x:xs) = minimum xs : map (min x) (otherMinimums xs)

color m V = [1..m]
color m (E t1 t2) = let
    c1 = color m t1
    c2 = color m t2 in
    zipWith (+) (otherMinimums c1) c2

t3 = E t2 $ E t2 $ E t2 $ E t2 $ t2
t2 = E t1 $ E t1 $ E t1 $ E t1 $ t1
t1 = E t0 $ E t0 $ E t0 $ E t0 $ t0
t0 = V

:

> color 3 t3
[157,158,163]
> color 4 t3
[157,158,159,156]