References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt145",{id:"formSmash:upper:j_idt145",widgetVar:"widget_formSmash_upper_j_idt145",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt146_j_idt148",{id:"formSmash:upper:j_idt146:j_idt148",widgetVar:"widget_formSmash_upper_j_idt146_j_idt148",target:"formSmash:upper:j_idt146:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

Limit laws for functions of fringe trees for binary search trees and random recursive treesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, 4Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2015. Vol. 20, 4
##### Keyword [en]

Fringe trees, Stein's method, Couplings, Limit laws, Binary search trees, Random recursive trees
##### National Category

Probability Theory and Statistics
##### Identifiers

URN: urn:nbn:se:uu:diva-248646DOI: 10.1214/EJP.v20-3627ISI: 000350286000001OAI: oai:DiVA.org:uu-248646DiVA: diva2:802290
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt384",{id:"formSmash:j_idt384",widgetVar:"widget_formSmash_j_idt384",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt390",{id:"formSmash:j_idt390",widgetVar:"widget_formSmash_j_idt390",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt396",{id:"formSmash:j_idt396",widgetVar:"widget_formSmash_j_idt396",multiple:true});
Available from: 2015-04-12 Created: 2015-04-06 Last updated: 2016-02-17Bibliographically approved

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings. As a consequence, we give simple new proofs of the fact that the number of fringe trees of size k = k(n) in the binary search tree or in the random recursive tree (of total size n) has an asymptotical Poisson distribution if k -> infinity, and that the distribution is asymptotically normal for k = o(root n). Furthermore, we prove similar results for the number of subtrees of size k with some required property P, e.g., the number of copies of a certain fixed subtree T. Using the Cramer-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of l-protected nodes in a binary search tree or in a random recursive tree.

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1123",{id:"formSmash:lower:j_idt1123",widgetVar:"widget_formSmash_lower_j_idt1123",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1124_j_idt1126",{id:"formSmash:lower:j_idt1124:j_idt1126",widgetVar:"widget_formSmash_lower_j_idt1124_j_idt1126",target:"formSmash:lower:j_idt1124:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});