Black Depth in Red-black Tree?

Wikipedia's Red-black tree states the last property of a Red-black tree:

Every path from a given node to any of its descendant NIL nodes contains the same number of black nodes. Some definitions: the number of black nodes from the root to a node is the node's black depth; the uniform number of black nodes in all paths from root to the leaves is called the black-height of the red–black tree

I'm not understanding this property. So, looking at this tree from the above Wikipedia article:

What is this field's value for the 8 tree, i.e. Root (13) -> 8?

How about for 15, i.e. Root (13) -> 7 -> 15?

When providing an answer, please also explain the why of that number.

• The definition you quote concerns "Every path from a given node to any of its descendant NIL nodes", but you ask about "this field's value" for paths 13 -> 8 and 13 -> 7 -> 15 that are not of such a form. I'm not sure what you are asking. Probably you meant 17 in place of 7. – hardmath Mar 6 '16 at 16:51

From the definitions:

The number of black nodes from the root to a node is the node's black depth.

Let's use $d(n)$ for the black depth of a node $n$. So $d(8) = 1$, for example, because one node is black along the path $13 \to 8$ (namely node $13$). Similarly $d(15)=2$ because along the path $13 \to 17 \to 15$, two nodes ($13$ and $15$) are black.

The uniform number of black nodes in all paths from root to the leaves is called the black-height of the red–black tree.

The black-height of the tree here is $3$ because $d(n)=3$ whenever $n$ is NIL.

In red-black tree you know: that black-depth is permanent for every two child in the tree. For example:

8:

We have two children's $1$ and $11$ and for them we know that black-depth($1$) = black-depth($11$)=$2$.