I am new to graph theory and need some help. I have an undirected tree with a few coloured nodes. My task is to calculate the minimum distance from each node to any of the coloured nodes in $$$O(N)$$$. N is the number of nodes. The number of coloured nodes may be up to N.

My solution: Initialise an array of size N with infinity as the elements, this will represent my min distance array, say $$$A$$$. Change $$$A[i] = 0$$$, if $$$i$$$ is a coloured node. Run a dfs from each coloured node and each recursive call of the dfs ends if it either reaches a leaf or a node with $$$A[i]$$$ less than or equal to that of the parent. Store the distances on A along the way.

But I suspect that the complexity of this approach is $$$O(N^2)$$$, consider a tree with a root and many children, and let these children have many more single child each, and let the leaves be coloured, kind of like a flower. Since I don't know of a problem that is based on this, I cannot verify anything.

If anyone can prove if it's $$$O(N)$$$ or $$$O(N^2)$$$ then please help me. If it has bad complexity or downright wrong then please help me with a solution.

Another approach I though of was to simultaneously run a dfs from all the coloured nodes and end a recursive call of each node if it's reached a previously reached node, kind of like a radiation from the coloured nodes. But I don't even know if that's possible for me to implement.

help pls