Black Depth in Red-black Tree?
$begingroup$
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.
trees data-structure
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
$endgroup$
add a comment |
$begingroup$
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.
trees data-structure
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
$endgroup$
1
$begingroup$
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 paths13 -> 8and13 -> 7 -> 15that are not of such a form. I'm not sure what you are asking. Probably you meant17in place of7.
$endgroup$
– hardmath
Mar 6 '16 at 16:51
add a comment |
$begingroup$
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.
trees data-structure
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
$endgroup$
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.
trees data-structure
trees data-structure
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
edited Jun 16 '17 at 14:10
Smylic
4,53921225
edited Jun 16 '17 at 14:10
Smylic
4,53921225
edited Jun 16 '17 at 14:10
Smylic
4,53921225
4,53921225
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
asked Mar 6 '16 at 16:36
Kevin MeredithKevin Meredith
281522
281522
1
$begingroup$
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 paths13 -> 8and13 -> 7 -> 15that are not of such a form. I'm not sure what you are asking. Probably you meant17in place of7.
$endgroup$
– hardmath
Mar 6 '16 at 16:51
add a comment |
1
$begingroup$
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 paths13 -> 8and13 -> 7 -> 15that are not of such a form. I'm not sure what you are asking. Probably you meant17in place of7.
$endgroup$
– hardmath
Mar 6 '16 at 16:51
1
1
$begingroup$
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.$endgroup$
– hardmath
Mar 6 '16 at 16:51
$begingroup$
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.$endgroup$
– hardmath
Mar 6 '16 at 16:51
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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($11$)=$2$.
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
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($11$)=$2$.
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
$endgroup$
add a comment |
$begingroup$
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($11$)=$2$.
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
$endgroup$
add a comment |
$begingroup$
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($11$)=$2$.
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
$endgroup$
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($11$)=$2$.
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
answered Mar 6 '16 at 16:48
openspaceopenspace
3,4452822
3,4452822
add a comment |
add a comment |
$begingroup$
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.
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
$endgroup$
add a comment |
$begingroup$
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.
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
$endgroup$
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.
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
edited Mar 6 '16 at 17:30
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
answered Mar 6 '16 at 17:19
ThéophileThéophile
19.7k12946
19.7k12946
add a comment |
add a comment |
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$begingroup$
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 -> 8and13 -> 7 -> 15that are not of such a form. I'm not sure what you are asking. Probably you meant17in place of7.$endgroup$
– hardmath
Mar 6 '16 at 16:51