Question

Please explain each answer

Problems Red-Black Trees

(a) Given that we follow the insertion operations; What is the largest possible number of red nodes in a red-black tree of n nodes? What is the smallest possible number?

(b) Can a black node have one red child and one black child ? If we relax the condition such that the children of a red node can be either red or black, what are operations necessary to keep the tree balanced ?

(c) Would the height and black height of the Red-Black Tree change if the same set of numbers are entered as a different sequence

(d) Create a Red-Black Tree, by inserting the numbers in the given sequence {8,7,5,3,2,6,4,1}

Answer 1

Dear student I understand you need help with some of the questions in red black tree.Please go through the following answers so that all the above queries will be resolved asap.

SO I'll start by answering subpart A of your question

The smallest number of internal nodes in a red-black tree with black height of k is 2^k-1

The largest number of internal nodes with black height of k is 2^2k-1.

Lets move ahead and I'll start answering subpart B in the easiest and the simplest way possible.

this cannot happen because no two consecutive nodes can be red. 2. If the child is black...

Naturally: Property IV guarantees that a Red-Black tree is adjusted in the event that it doesn't contain red hubs, since each root-leaf way has similar number of dark hubs. At the point when red hubs are included, Property III guarantees that, on a root-to-leaf way with k dark hubs, there are all things considered k red hubs.

Lets start answering Subpart C so that you have a better understanding of this topic pal.

A red–dark tree is a sort of self-adjusting parallel inquiry tree. Every hub stores an additional piece speaking to shading, used to guarantee that the tree remains around adjusted during inclusions and deletions.

At the point when the tree is adjusted, the new tree is modified and repainted to reestablish the shading properties that compel how unequal the tree can become in the most pessimistic scenario. The properties are planned with the end goal that this adjusting and recoloring can be performed productively.

The re-adjusting isn't great, however ensures looking in O(log n) time, where n is the quantity of hubs of the tree. The inclusion and cancellation tasks, alongside the tree revamp and recoloring, are likewise acted in O(log n) time.

Following the shade of every hub requires just 1 bit of data for each hub on the grounds that there are just two tones. The tree doesn't contain some other information explicit to its being a red–dark tree so its memory impression is practically indistinguishable from a work of art (uncolored) paired inquiry tree. By and large, the extra digit of data can be put away at no extra memory cost.

Now I am providing you with the red black tree for the given sequence {8,7,5,3,2,6,4,1} and answering the last and final subpart D.Please go through the attached Image as well as all the information above.Hope all your questions have been answered and you have a thorough understanding of the topic.