Minimum ipl(n) and epl(n) in any BS tree on n nodes

A binary tree T on n nodes has the minimum internal path length among all binary trees on n nodes,

that is,

ipl(T)=(Subscript[ipl, min])(n)

iff

T is empty (so n = 0)

or

all levels of T, except perhaps for the last level of T, have the maximum number of nodes.

Recall that the maximum number of nodes in level k of a binary tree is (2^k).

Here is example of such a BS tree on nodes 1 through 11:

Hence, the minimum internal path length 
(Subscript[ipl, min])(n)is equal to the internal path length
ipl(H) in a heap on n nodes.

Thus,

(Subscript[ipl, min])(n)=(∑_(i=1)^n)⌊Log2[i]⌋=
(n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2=
=(n+1) (Log2[n+1]+ϵ)-2 n,

where ϵ is a function of n+1 oscillating between:
0≤ϵ< 0.086071332056.

Note. 
(n+1) (Log2[n+1]+ϵ)-2 n 
is the number of comparisons of keys that Quicksort will perform in the best case while sorting an n-element array with no duplicate keys.

Plot[(n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2,{n,1,20}]

So,
(Subscript[ipl, min])(n)≈(n+1) Log2[n]-2 n.

Plot[{(n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2,(n+1) Log2[n]-2 n},{n,1,50}]

Therefore,

(Subscript[epl, min])(n)=(Subscript[ipl, min])(n)+2n≈(n+1) Log2[n]

or

(Subscript[epl, min])(n)≈(n+1) Log2[n]

Average number of comparisons for successful search in an average BS tree:

(Subscript[c, n])=(((Subscript[ipl, avg])[n])/n)+1≈ (((n+1) Log2[n]-2 n)/n)+1≈
Log2[n]-2+1 = Log2[n]-1.

(Subscript[c, n])≈Log2[n]-1

Plot[{1+((n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2)/n,Log2[n]-1},{n,1,200},PlotPoints-> 200]

Here is a bit more accurate approximation:

(Subscript[c, n])≈Log2[n]-1+((Log2[n])/n)

Plot[{1+((n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2)/n,Log2[n]-1+((Log2[n])/n)},{n,1,200},PlotPoints-> 200]

Average number of comparisons for unsuccessful search in an average BS tree:

(Subscript[c, n]^('))=(((Subscript[epl, avg])[n])/(n+1))≈ (((n+1) Log2[n])/(n+1))=Log2[n]
1.386 Log2[n]-0.846

(Subscript[c, n]^('))≈Log2[n].

Plot[{((n+1)⌊Log2[n+1]⌋-(2^(⌊Log2[n+1]⌋+1))+2+2n)/(n+1),Log2[n]},{n,1,200},PlotPoints-> 200]

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