B+ tree deletion

Question

In this by deleting 298 last node is not staifying half filled property so 65 is adjusted in that last node 

By this now tree becomes as

                                  [|45|]
                            [|5|29|]   [|51|59|65|]
[|1|3|4|] [|5|9|] [|29|37|43|] [|45|47|49|] [|51|53|57|] [|59|61|] [|65|165|]

every node is ok then why 29 is made as root

Don't whether my doubt is correct 

can anybody say please

Comments

http://sketchingdream.com/blog/b-plus-tree-tutorial/ 

I refered

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if d = 4, then,

In this example on deleting 45 why the internal node 

[|45|51|59|65|]

is changed to 

[|43|51|59|65|]

I think it should be 

[|47|51|59|65|]

Since in the given example they are borrowing key from left sibling even after deleting we have 2 keys left in the leaf node, which is >= ceil(d/2).

Since if the no of key values is below ceil(d/2) then only  we can take borrow.

Answer 1

There is no need to change the root node for deleting the leaf node

since by deleting 298 there is underflow so 65 in left sibling is transferred to the neighboring node and search key should be updated

so the search key node will be 51 59 65.

your explanation is correct.....

Comments

One more thing if order of tree is 3 then

no. of keys in node=ceil(3/2)-1 =1

So node should have minimum 1 key

if order of tree is 4 then

no. of keys in node=ceil(4/2)-1 =1

So node should have minimum 1 key

Is this not correct ??

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Yes correct

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I have seen in some sites

Minimum number of keys=floor(n/2) where n is order  ---------------- 1

and

in other

Minimum number of children =ceil(n/2) and keys=ceil(n/2)-1   -------------- 2

if n=4

for 1 minimum keys=2

for 2 minimum keys=1

which one is correct which to follow