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.
trees data-structure
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin Meredith
281522
add a comment |
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 Meredith
281522
1
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.
– hardmath
Mar 6 '16 at 16:51
add a comment |
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 Meredith
281522
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 Meredith
281522
edited Jun 16 '17 at 14:10
Smylic
4,53921225
asked Mar 6 '16 at 16:36
Kevin 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 Meredith
281522
asked Mar 6 '16 at 16:36
Kevin Meredith
281522
asked Mar 6 '16 at 16:36
Kevin Meredith
281522
281522
1
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.
– hardmath
Mar 6 '16 at 16:51
add a comment |
1
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.
– hardmath
Mar 6 '16 at 16:51
1
1
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
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
add a comment |
2 Answers
2
active
oldest
votes
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
openspace
3,4452822
add a comment |
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
Théophile
19.5k12946
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1685705%2fblack-depth-in-red-black-tree%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
openspace
3,4452822
add a comment |
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
openspace
3,4452822
add a comment |
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
openspace
3,4452822
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
openspace
3,4452822
answered Mar 6 '16 at 16:48
openspace
3,4452822
answered Mar 6 '16 at 16:48
openspace
3,4452822
answered Mar 6 '16 at 16:48
openspace
3,4452822
3,4452822
add a comment |
add a comment |
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
Théophile
19.5k12946
add a comment |
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
Théophile
19.5k12946
add a comment |
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
Théophile
19.5k12946
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
Théophile
19.5k12946
edited Mar 6 '16 at 17:30
answered Mar 6 '16 at 17:19
Théophile
19.5k12946
answered Mar 6 '16 at 17:19
Théophile
19.5k12946
answered Mar 6 '16 at 17:19
Théophile
19.5k12946
19.5k12946
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1685705%2fblack-depth-in-red-black-tree%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
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.– hardmath
Mar 6 '16 at 16:51