GATE CSE
First time here? Checkout the FAQ!
x
+3 votes
196 views
A $3-\text{ary}$ tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a $3-\text{ary}$ tree with $n$ internal nodes is $2(n-1)$.
asked in DS by Veteran (58.4k points)   | 196 views

I think in the question, number of leave nodes(L) for n internal nodes in a 3 ary tree is

 L= 2(n-1) + 3 = 2n+1

1 Answer

+4 votes
No of nodes at level $i =3^i$

Let height of tree be $h$

So total no of internal nodes $=3^0+3^1+3^2+\dots +3^{h-1}=\frac{3^{h}-1}{2}$

$2n=3^{h}-1$

No of leaf nodes $=3^h =2n+1=2(n-1)+3$

Let us prove by induction

Base case

$n=1$ (one internal node i.e., root node)

No of leaves $=2(1-1)+3=3$

Let it be true for $n$ internal nodes

Now we prove for $m$ nodes where $m=n+1$

We have $L(m)=2(n+1-1)+3$

Also $L(m)=L(n)+3-1\\=2(n-1)+3+3-1\\=2n+3$

So if $L(n)$ is true then $L(n+1)$ is also true

Hence proved by induction.
answered by Veteran (31k points)  
edited by

Apart from proving it by induction 

I was wondering does this formula hold correct ?

eg 

n = 2 L= 5

but by L = 2(n-1) = 1 ???????? How is it correct

let there be n internal nodes (including root)  and L leaves

then using degree 

(n-1)4 + 3 + L = 2( n+L-1)

L= 2n + 1

eg in above pic n = 2 so L = 5

(Just trying to learn calculating no of leaves - not worrying about proving by induction .....just want to know if formula is correct ? )

As per the wordings given in question your tree is correct. But i guess they meant a full ternary tree- otherwise no. of leaves cannot be a fixed value.
Can u plz draw a tree @arjun sir where the question's no. of leaf requirement will satisfy?

I m not getting any..
@Arjun ...i think question is wrong


Top Users Apr 2017
  1. akash.dinkar12

    3660 Points

  2. Divya Bharti

    2580 Points

  3. Deepthi_ts

    2040 Points

  4. rude

    1966 Points

  5. Tesla!

    1768 Points

  6. Debashish Deka

    1614 Points

  7. Shubham Sharma 2

    1610 Points

  8. Prashant.

    1492 Points

  9. Arjun

    1472 Points

  10. Arunav Khare

    1464 Points

Monthly Topper: Rs. 500 gift card

22,088 questions
28,063 answers
63,298 comments
24,173 users