If the tree has n
nodes and the root is black, there are n - 1
nodes left and n - 1 = O(n)
, so you're right.
If you want to count/bound number of red nodes in the tree more accurately, you have to know the topology of that tree.
For instance, if the tree is a complete binary tree, according to the definition of the red-black-tree, it can have no red nodes at all.