data structures - For "any" binary tree, what is the relation between all nodes and internal nodes -


the original question below.

for any binary tree n nodes , internal nodes, relation between n , _____ <= i.

in opinion

i thought n/2 <= i, can't not figure out condition let n/2 = i. want ask root node internal node?

in full tree there 1 + 2 + ... + 2^(h-1) internal nodes , 2^h leaf nodes (where h height of tree).

that means total number of nodes 

n = 1 + 2 + .... + 2^h = 2^(h+1)-1 = 1 + 2 + ... + 2^h-1 = 2^h - 1

now, relation looking for:

n-i = 2^h+1 - 1 - 2^h + 1 = 2^h+1 - 2^h = 2*2^h - 2^h = 2^h

since 2^h = i+1, get:

n - = i+1 n - 1 = 2i (n - 1)/2 =

(note there no issue non integer result, since n odd in full tree).

now, have left show full tree indeed 1 highest such ratio (this left reader).


Comments

Post a Comment

Popular posts from this blog

Is there a better way to structure post methods in Class Based Views -

i'm using class based views in django , never specify url in ajax calls , omit action parameter in form because know these requests go through post method of class based view. notice view - profile page make many requests depending on user updating on page - post method become convoluted , if/else mess example: def post(self, request, *args, **kwargs): # handle user changing available date if request.post.get('availabledate') != none: self.updateavailabledate(request) return jsonresponse({'result':'success'}) if request.post.get('newprojectname') != none: creative_user = creativeuserprofile.objects.get(id = request.user.id) project = project.create_project(creative_user, request.post['newprojectname']) project.save() return jsonresponse({'projectid': project.id }) if request.post.get('projectimage') != none , request.files['file']: proj
Read more

performance - Why is XCHG reg, reg a 3 micro-op instruction on modern Intel architectures? -

i'm doing micro-optimization on performance critical part of code , came across sequence of instructions (in at&t syntax): add %rax, %rbx mov %rdx, %rax mov %rbx, %rdx i thought had use case xchg allow me shave instruction , write: add %rbx, %rax xchg %rax, %rdx however, dimay found agner fog's instruction tables , xchg 3 micro-op instruction 2 cycle latency on sandy bridge, ivy bridge, broadwell, haswell , skylake. 3 whole micro-ops , 2 cycles of latency! 3 micro-ops throws off 4-1-1-1 cadence , 2 cycle latency makes worse original in best case since last 2 instructions in original might execute in parallel. now... cpu might breaking instruction micro-ops equivalent to: mov %rax, %tmp mov %rdx, %rax mov %tmp, %rdx where tmp anonymous internal register , suppose last 2 micro-ops run in parallel latency 2 cycles. given register renaming occurs on these micro-architectures, though, doesn't make sense me done way. why wouldn't register renamer
Read more

jquery - Responsive Navbar with Sub Navbar -

good day, forgive me nav vocabulary not best. if question confusing apologize. i'm working on navigation bar needs centered on page @ 768px , above. items in nav open (or perhaps toggle) subnav lives directly below. these nav should appear centered on page. i have been able work through responsive portion of nav bar, having main items align vertically , sub nav menu of each item roll out below, , gave them indent separation. working well. however, issue not responsive side, behavior @ width greater 768px. happens toggling menu's sub navigation pushes other main items down page below sub nav. here's code: $(document).ready(function () { //stick in fixed 100% height behind navbar don't wrap $('#slide-nav.navbar-inverse').after($('<div class="inverse" id="navbar-height-col"></div>')); $('#slide-nav.navbar-default').after($('<div id="navbar-height-col"></d
Read more