trees and binary trees

[elixyre]

Hi,

I propose several methods for trees and binary trees:

File : trees_classicals_algorithms_EliX-jbp: several classical operations on trees and binary trees

• on abstract trees:
  • depth pre/post order transversal algorithm
  • breadth first order transversal algorithm
• on binary trees:
  • infix order transversal algorithm
  • canonical permutation associated to the left/right binary search tree insertion
  • left/right rotate (for labelled and unlabelled BT)

File: trees_research_algorithms_EliX-jbp: several research algorithms on binary trees

• over/under (Loday-Ronco)
• pred/succ in the Tamari lattice
• hook length formula

File: trees_latex_output_EliX-jbp: a method for nice latex output

Apply:

• trac_14498-algorithms_trees-rebased.patch​
• trac_14498-tree-imps-dg-rebased.patch​
• trac_14998-more-edits-dg.patch​
• trac_14498-further-docfixes-dg.patch​

[chapoton]

Maybe there is a dependency on #8703 ?

[elixyre]

Replying to chapoton:

Maybe there is a dependency on #8703 ?

Yes that's true.

[darij]

A nice step towards the Hopf algebras. Some comments:

@Classical algorithms:

Typo appearing twice: "explorer" (should be "explores"). Also, "An other" -> "Another".

Not sure, but I also think "transversal" should be "traversal".

The docstrings fail to explain an important point: what exactly "manipulate" means (and, correspondingly, what the "action" variable is for). The first time I read them I thought the methods output the list of nodes in the respective order! The doc for in_order_transversal should explain the difference between node_action and leaf_action. By the way, why do the other methods have only 1 type of action?

The example for in_order_transversal has two different things called "b". Not a big issue, of course.

I don't understand what "the canonical permutation associated to the binary search tree insertion" is supposed to mean; is this a notation from one of Loday(-Ronco)'s papers?

Copypaste error: the docstring for left_rotate says "Right". (Both times.)

@Research algorithms:

Is computing the hook_length_formula by symbolic integration really easier than just recursively multiplying the hook_length_formulas for the left and right subtrees and then multiplying by an appropriate binomial coefficient? I'm not saying it isn't, just asking.

[chapoton]

Some doctests are failing, please correct them.

[elixyre]

This new patch contains all the modifications.

Replying to darij:

Typo appearing twice: "explorer" (should be "explores"). Also, "An other" -> "Another".

Not sure, but I also think "transversal" should be "traversal".

Typo ok?

The docstrings fail to explain an important point: what exactly "manipulate" means (and, correspondingly, what the "action" variable is for). The first time I read them I thought the methods output the list of nodes in the respective order! The doc for in_order_transversal should explain the difference between node_action and leaf_action. By the way, why do the other methods have only 1 type of action?

I tried to explain what is "action". But my english is very bad so... About, action and leaf/node_action... The distinction between leaf and node on Abstract Trees class is not clear for me.

I don't understand what "the canonical permutation associated to the binary search tree insertion" is supposed to mean; is this a notation from one of Loday(-Ronco)'s papers?

The term canonical is may be not a good choose, this method is suppose to compute a representant of the sylvester class...

Copypaste error: the docstring for left_rotate says "Right". (Both times.)

OK

@Research algorithms:

Is computing the hook_length_formula by symbolic integration really easier than just recursively multiplying the hook_length_formulas for the left and right subtrees and then multiplying by an appropriate binomial coefficient? I'm not saying it isn't, just asking.

I implement the q_hook_length_formula.

Replying to chapoton:

Some doctests are failing, please correct them.

OK

[VivianePons]

Hi,

I cannot apply your patch, maybe it's conflicting with #14731 and #14784 which have already been positively reviewed are already or will be merged in sage 5.11. Also, we actually wrote twice the same things, I have two methods "to_132_avoiding_permutation" and "to_312_avoiding_permutation" which are somehow similar to your "canonical_permutation". I think we should keep my name which is clearer. But for these methods, I wrote a "_postfix_word" method which is similar to your "post_order_traversal" thing (but I think yours is better and with better documentation). But my patch has already been merged, so you should write yours on top of mine and just replace my methods by yours when they're better.

About names, I think "sylvestrohedron_greater" is not a good name, we should say "Tamari_greater", because it's more commonly used.

Do you want to come to Marne-la-vallee so that we can work on the merge of your patch with mine? I'll be there Thursday, Friday and Monday. Then again at the end of July. Please tell me

[elixyre]

Hi,

I have removed canonical permutation and add some other classical methods on trees (is_perfect, complete, ...).

I don't modify your "_postfix_word" with "post_order_traversal" because it is more specific... in the traversal algorithm we don't care about the children order.

The "sylvestrohedron" is become "tamari".

Thanks, JB

[chapoton]

please write which patches should be applied

• in a comment, for the bot
• in the description, for humans

[darij]

Hi Jean-Baptiste, thanks for the changes (but aren't the "An other" and "explorer" typos still there?).

In the q-hook-length formula, are you sure you want to work in the symbolic ring? That is, you really want not to do any cancellations? For example, your doctested example

            sage: BinaryTree([[],[]]).q_hook_length_formula()
            (((q + 2)*q + 2)*q + 1)*q/((q + 1)*q + 1)

simplifies to q^2 + q. I have not done any speed tests, but I would be surprised if the fraction field of a univariate polynomial ring wasn't faster than the symbolic ring (also I don't trust the symbolic ring, but this might be unfounded). Alternatively, you can make a recursive computation with q-binomial coefficients that never leaves the polynomial ring; but not sure if this is very efficient (q-binomial coefficient might be dense).

I think the formula in the docstring of q_hook_length_formula() doesn't match the algorithm. In the formula, there is a positive power of q in the denominator (which should give a Laurent polynomial, not a polynomial), while in the algorithm there is a negative power of q in the denominator (which gives a polynomial divisible by some power of q in the end). I'm not sure why the power of q is there at all...

I hate to say I am still confused by the input syntax of trees:

sage: b = BinaryTree([[None,[]],[[[],[]],[]]]).canonical_labelling()   
sage: parent(b)
Labelled binary trees
sage: b
3[1[., 2[., .]], 7[5[4[., .], 6[., .]], 8[., .]]]
sage: OrderedTree(b)
[[[], [[], []]], [[[[], []], [[], []]], [[], []]]]
sage: b.node_number()
8
sage: OrderedTree(b).node_number()
17

Apparently the coercion(!) from b to an ordered tree replaces every leaf by a node with two children because for some reason leaves of binary trees are stored as nodes-with-two-children internally. This is all defensible from some viewpoint, but I want it doced. Once somebody clears this up I would volunteer to review #14498, but so far I don't want to spread my own confusion...

[darij]

not a review but just a fix for a couple typos. Jean-Baptiste, can you please look this over and qfold into your patch?

[elixyre]

Hello,

I added your modifications and I modify *q_hook_length_formula* with ... I am not sure about what to do with that... could you check it is correct now.

Replying to chapoton:

please write which patches should be applied

• in a comment, for the bot
• in the description, for humans

For human, apply: trac_14498_algorithms_trees_13_07_15_EliX-jbp.patch

For the bot... I don't know how to do a comment.

Thanks, Jean-Baptiste

[chapoton]

well, what you have just written is a comment. And what I write now is another one.

for humans, I have added one line in the description (see top of the page)

for the bot, you just have to write the following line

apply trac_14498_algorithms_trees_13_07_15_EliX-jbp.patch

[chapoton]

the patch does not apply on 5.11.beta3

[elixyre]

Hi every one,

I don't understand why the patchbot don't work... I try (again) on my 5.11beta3 and all seems to be for me... Someone can looks that??

Thanks,

Jean-Baptiste Priez

[chapoton]

Have you made your patch using "hg export" or in some other strange way (by hand) ?

The standard process is to use

hg export tip > $HOME/trac_xxx.patch

to export the patch as a file, after making sure your patch is the top applied patch. Then upload the resulting file to trac.

Any other way is bound to have strange results..

It looks like you have concatenated several patches by hand maybe ? You must use hg qfold to do that.

---

Quick comments on the code:

it is a bad idea to use

q = SymbolicRing().var('q')

to have a polynomial variable. Much better is

q = polygen(QQ, 'q')

In the same hook-length function, you should write "There are 20 permutations"

[elixyre]

Hello,

Replying to chapoton:

Have you made your patch using "hg export" or in some other strange way (by hand) ?

The standard process is to use

hg export tip > $HOME/trac_xxx.patch

to export the patch as a file, after making sure your patch is the top applied patch. Then upload the resulting file to trac.

I use

sage: hg_sage.apply(...)

Any other way is bound to have strange results..

I will try to use hg export...

It looks like you have concatenated several patches by hand maybe ? You must use hg qfold to do that.

---

Quick comments on the code:

it is a bad idea to use

q = SymbolicRing().var('q')

to have a polynomial variable. Much better is

q = polygen(QQ, 'q')

In the same hook-length function, you should write "There are 20 permutations"

Thanks for comment, i will modify that.

[chapoton]

the patch does not applies on 5.12.beta3

as long as the patch is not clean, this cannot be considered as "needs review"

[elixyre]

Hello,

May be, I understand nothing but my patch apply in sage 5.12.beta5.

If there is a problem, it doesn't appear on my laptop. (Sage on Mac os is more clever than on linux!)

So if the patch does not applies again, please correct that because I don't have that problem... when that thing will be solve I will modify _q = SymbolicRing?.._ if you don't do it but now I don't want spend 3hours to make a new patch (which doesn't applies) for one simple line...

[tscrim]

Hey Jean-Baptiste,

The patch applies for me on 5.12.beta5 (in the combinat queue), but there doesn't seem to be anything that could be a dependency. My guess is the patched together patch (sorry, couldn't resist the pun) is why it's not applying, and I agree that this needs to be fixed.

Also, because of the accented character, you need to add

# -*- coding: utf-8 -*-

as the first line of binary_tree.py (I put a small patch in the combinat queue which does this).

Thanks,
Travis

[chapoton]

Hello Travis,

maybe you can backport the patch from the sage-combinat queue to trac (folded with your small patch) ?

Frederic

[tscrim]

Hey everyone,

Here's a new version with my patch small header patch folded in, the patch file cleaned up, and my docstring tweaks (I also marked some tests as # long time).

Best,
Travis

For patchbot:

Apply: trac_14498-algorithms_trees-rebased.patch​

[chapoton]

apply trac_14498-algorithms_trees-rebased.patch

[darij]

I have just uploaded trac_14498-tree-imps-dg.patch​ (to be applied after trac_14498-algorithms_trees-rebased.patch​) which should fix #15060 and clear up some of the vertex-vs.-node confusion on binary trees. Can anyone look over it and tell me if I got things right? I cannot really start reviewing this patch before this is sorted out.

[tscrim]

minor docstring fix

[darij]

ignore this file

[darij]

I have renamed and mostly redone the q-hook length formula method. Elixyre, is it still what you wanted? Among other changes,

• the default option now has no q-factor,
  
• the q-factor is determined by the right child (as the doc said) rather than the left one (as the earlier code did),
  
• the computation is being done in whatever ring the user is given, rather than the symbolic ring.

[darij]

improvements and bugfixes

[darij]

I have posted a new version of trac_14498-tree-imps-dg.patch​. Unfortunately a lot more is needed for a review, but I think this is a step forward. The following issues have been fixed:

• The issues discussed in #14564 (leaves vs. leaves-and-nodes inconsistencies, and border cases) have been fixed.
  
• The implementation and the documentation of the q-hook-length formula has been revamped (even the name is now different).
  
• Various documentation has been rewritten.

Things that remain to be done:

• BinaryTree() is a functioning shortcut for BinaryTree(None), but LabelledBinaryTree() is not a functioning shortcut for LabelledBinaryTree(None). Should this be fixed? Using init_extra?
  
• It is not obvious to me that the tamari_succ and tamari_prec methods actually compute the successors resp. predecessors in the Tamari order. Is there a proof somewhere in the literature? It's clear that every successor of a tree t in the Tamari poset is obtained by a right rotation, but is it clear that every right rotate is actually a successor?
  
• The over method defines the "over" operation. As I wrote in the docstring: " Fix this doc and figure out which of the different (inconsistent?) notations to follow. I don't see the "over" operation defined in [LodayRonco?]_, although [HNT05]_ pretends that it is defined there. In [HNT05]_ the definition is somewhat vague about the order of the operads. Loday and Ronco seem to define "over" on page 2 of "Order structure on the algebra of permutations and of planar binary trees", but with a different choice of left/right."
  
• The docstring of the class LabelledBinaryTree could be less barren; in particular it could explain how to initialize these.
  
• The binary_search_insert docstring might use a definition of binary search insertion, given that most sources only do it in the standard (i. e., no two equal labels) case.
  

I fear I won't come back to this patch very soon, so everyone who feels like improving these is invited to do so.

For the patchbot:

apply trac_14498-algorithms_trees-rebased.patch​ trac_14498-tree-imps-dg.patch​

[chapoton]

patchbot apply trac_14498-algorithms_trees-rebased.patch​,trac_14498-tree-imps-dg.patch

[chapoton]

apply trac_14498-algorithms_trees-rebased.patch​ trac_14498-tree-imps-dg.patch

[chapoton]

apply trac_14498-algorithms_trees-rebased.patch​ trac_14498-tree-imps-dg.patch

[chapoton]

apply trac_14498-algorithms_trees-rebased.patch trac_14498-tree-imps-dg.patch

[chapoton]

Darij, I have rebased your patch, it was not applying cleanly.

patchbots, please

apply trac_14498-algorithms_trees-rebased.patch trac_14498-tree-imps-dg-rebased.patch

[chapoton]

new patch, correcting the failing doctest

[darij]

Frederic, I think that my patch was supposed to apply atop of trac_14498-algorithms_trees-rebased.2.patch , not atop of trac_14498-algorithms_trees-rebased.patch. Is that the reason for the unclean apply?

EDIT: OOPS, I am just seeing that it was me who fucked up the apply declaration. Sorry!

It should be

apply trac_14498-algorithms_trees-rebased.2.patch trac_14498-tree-imps-dg.patch

Is there a chance you could rebase your changes back? Thanks a lot. (A qfold would probably be best. It seems that once a something.2.patch file appears, chaos is bound to happen.)

[chapoton]

Hello,

well, I would rather not change anything back. It was a very simple rebase, and I am rather sure there is nothing lost in your patch. I have checked that nothing has got lost in Travis patch, too.

So let us stay with

apply trac_14498-algorithms_trees-rebased.patch trac_14498-tree-imps-dg-rebased.patch

[chapoton]

I have now made further corrections to the documentation.

[darij]

Hi Federic,

I didn't want to say anything got lost in *my* patch. It's Travis' patch where things could have gotten lost. But I'm seeing that you re-did all of the edits present in trac_14498-algorithms_trees-rebased.2.patch but absent from trac_14498-algorithms_trees-rebased.patch. So nothing got lost. Sorry again for the trouble!

Darij

[tscrim]

The non-ascii character should be okay since the corresponding file has UTF-8 encoding. I'm thinking the startup time increase is due to setting up the latex stuff in abstract_tee.py and that probably should be done in the _latex_() method the first time it is called. Any objections to moving it there?

Best,
Travis

[darij]

qfold -- used (mostly by myself) for more comfortable viewing

[darij]

new review patch

[darij]

I've completed my review of this patch, but due to its size (and its introduction of another method for my own research) someone will have to review my review. Frederic?

Thanks Viviane for the help with Loday's over/under operations. I still can't find them in the Loday-Ronco paper, but at least I now know their definitions!

Yes, my patch applies *on top* of the algorithms and tree-imps patches. Sorry for the typo in the trac number.

for the patchbot:

apply trac_14498-algorithms_trees-rebased.patch​ trac_14498-tree-imps-dg-rebased.patch trac_14998-more-edits-dg.patch​

[darij]

Sorry guys. Had to introduce further docfixes after noticing that several methods fail to preserves labels on leaves of binary trees. The whole issue of leaves not being fully-accepted vertices of binary trees is extremely confusing...

for the patchbot:

apply trac_14498-algorithms_trees-rebased.patch​ trac_14498-tree-imps-dg-rebased.patch trac_14998-more-edits-dg.patch​ trac_14498-further-docfixes-dg.patch​

[darij]

now with fixed doc :)