Counting labels in binary trees

We investigate properties of binary (search) trees related to the inorder labelling of the nodes. Both permutation and uniform trees are considered. We give explicit formulas to count the number of interior, middle and final nodes (leaves) containing a specific label. Possible applications are discussed.This work was supported by the Italian Ministry for Public Education.     

Self-assembled monolayers and electrostatically self-assembled multilayers of polyalkylviologens on sulfonate-modified gold and indium–tin–oxide electrodes

Self-assembly (SA) of polyviologens with linear alkyl spacers PVn (n = 3–10) on sulfonate-primed gold and ITO electrode surfaces has been investigated by CV, QCM, XPS and AFM. The polymers form stable and relatively dense monolayers even from 10⁻⁴ M PVn concentration. The viologen surface density is decreased as n is increased but for n = 10 and on alkylsulfonate-primed surfaces a strong adsorption takes place, with decreased redox potential and very narrow cyclic voltammogram of the absorbed layer due to organized structures on the surface.Stable and regular electrostatically self-assembled (ESA) multilayers are built on sulfonate-primed surfaces with PV3 and polysulfonates. From XPS analysis, the composition of the PVn/polysulfonate multilayers corresponds to a 1:1 ratio of polyanion and polycation charges, i.e., no extra non-polymeric ion is present, independently from the alkane chain length. The rate of electron transfer within the multilayers (diffusion coefficient = ca. 10⁻⁹ cm² s⁻¹) is higher than for analogous bulk materials.     

New synthesis of optically active 5-isoxazolidinones and beta-amino acids

A new simple and stereoselective synthesis of 5-isoxazolidinones based on the reaction of lithiated 2-isopropyl-2-oxazolines with nitrones is described. A chiral version of such a methodology allows the preparation of highly enantioenriched 5-isoxazolidinones which are useful precursors for the synthesis of beta-amino acids     

Intramolecular Dynamics of Tetranuclear Carbonyliridium Cluster Compounds. Part II. Disubstituted complexes with one chelating ligand. Crystal structure of [Ir4(CO)7(μ-CO)3(1,5-cyclooctadiene)]

The two complexes [Ir₄(CO)₁₀(diarsine)] (1) and [Ir₄(CO)₁₀(1,5-cyclooctadiene)] (2) bear a bidentate ligand chelating one metal atom of the basal face of the Ir₄ tetrahedron. However, they differ in fluxional behaviour as observed by 2D-exchange and variable-temperature ¹³C-NMR. The CO-site exchange with lowest activation energy proceeds via an unbridged intermediate in 2 , whereas that in 1 occurs via a concerted edge-bridging of CO's to an alternative face of the metal core. This difference is apparently related to different ground-state geometries: the basal CO's are symmetrically bridging in 2 , whereas two CO's are semi-bridging in 1 . The molecular structure of 2 was determined by single crystal X-ray diffractometry. The crystals are orthorhombic, space group Pbca, a = 11.651(4), b = 13.118(3), c = 28.64(1)Å. The idealized molecular symmetry is C_s. The diolefin chelates a basal Ir-atom replacing an axial and a radial CO group on the tetrahedral metal-atom framework.     

Oxazolinyloxiranyllithium-mediated stereoselective synthesis of alpha-epoxy-beta-amino acids

[reaction: see text] The stereoselective synthesis of novel alpha-epoxy-beta-amino acids is described by a route that combines the chemistry of oxazolinyloxiranyllithiums with that of nitrones. The intermediate trioxadiazadispiro[2.0.4.3]undecanes 4 have been isolated and converted by hydrolysis into epoxy-5-isoxazolidinones 5 which can be transformed into the alpha-epoxy-beta-amino acids 8 by N-O reduction.     

Synthesis of the two enantiomers of the sex pheromone of Diabrotica Undecimpunct at a Howardi and of chiral precursors of other pheromones starting from enantiomerically pure methyl hydrogen (R)-3-methylglutarate

Readily available methyl hydrogen (R)-3-methylglutarate(2) is a useful chiral building block for the synthesis of several biologically active compounds. Enantiomerically pure (R)-2 has been employed to synthesize stereospecifically each of the two enantiomers, 1a and 1b, of 10-methyl-2-tridecanone, the sex pheromone of the southern corn rootworm, Diabrotica undecimpunctata howardi Barber. Compound (R)-2 has been also used to prepare 99% optically pure (R)-3-methyl-1-pentanol (6) and enantiomerically pure (R)-5-methyl-i-tricosyne (7). These compounds are useful building blocks suitable for the further elaboration to other chiral insect pheromones.     

An efficient route to tetrahydronaphthols via addition of ortho-lithiated stilbene oxides to alpha,beta-unsaturated fischer carbene Complexes

[reaction: see text] A stereoselective/stereospecific synthesis of polysubstituted tetrahydronaphthols based on the Michael addition of ortho-lithiated stilbene oxides to alpha,beta-unsaturated Fischer carbene complexes followed by an unusual cyclization of the corresponding intermediate in a 6-endo-tet mode is described.     

Positive chemical ionization and tandem mass spectrometric fragmentation for the gas chromatographic analysis of beta-agonists using the ion trap technique

A procedure is described for the determination of three characteristic beta-agonists (clenbuterol, terbutalin and salbutamol) based on the formation of the corresponding protonated molecules and related collisional experiments. Quantification was carried out on selected collisional fragments and the reproducibility of the relative abundances of these fragments was estimated. The performance of the method was tested on bovine urine samples spiked at the lowest level of 0.2 ng ml(-1) in each of the chosen compounds.     

Amide (A)–thioamide (T) interconversions using Ph3SiSH (A to T) and Ph3SnOH (T to A) reagents

Ph₃SiSH transforms amides to thioamides and Ph₃SnOH performs the reverse process, with the concomitant formation of Ph₃SiOH (or Ph₃SiOSiPh₃) and Ph₃SnSSnPh₃, respectively. The chemistry is a delightful illustration of the oxophilicity of silicon compared to the thiophilicity of tin and occurs under relatively mild conditions, and for amide to thioamide transformation requires no amide activation. The chemistry is in accord with available data for Si? (S)(O), Sn? (O)(S) and C?(O)(S) bond energies. Copyright © 2016 John Wiley & Sons, Ltd.