Allotropes of phosphorus Allotropes of phosphorus Elemental phosphorus can exist in several allotropes, the most common of which are white and red solids. Solid violet and black allotropes are also known. Gaseous phosphorus exists as diphosphorus and atomic phosphorus. White phosphorus, yellow phosphorus or simply tetraphosphorus (P) exists as molecules made up of four atoms in a tetrahedral structure. The tetrahedral arrangement results in ring strain and instability. The molecule is described as consisting of six single P–P bonds. Two different crystalline forms are known. The α form is defined as the standard state of the element, but is actually metastable under standard
Red–black tree has at least: internal nodes. Using this lemma we can now show that the height of the tree is logarithmic. Since at least half of the nodes on any path from the root to a leaf are black (property 4 of a red–black tree), the black-height of the root is at least h(root)/2. By the lemma we get: Therefore, the height of the root is . In addition to the single-element insert, delete and lookup operations, several set operations have been defined on red-black trees: union, intersection and set difference. Then fast "bulk" operations on insertions or deletions can be