A brief review of the nearly half century of research related to taper equations in the Faculty of Forestry at the University of British Columbia is presented. Two new variable-exponent taper models, the 2001 and 2002 models, are introduced and compared to Kozak's 1988 and models. This comparison, based on 38 species groups consisting of 53 603 trees, demonstrated that the 2002 model is consistently the best overall model of the four, and the 2001 model is the simplest in form and the best for estimating merchantable height. The results of this study also indicate that several fit statistics and lack-of-fit statistics should be used to create indices for ranking taper models for practical applications, instead of only a few.

A different approach to fitting taper equations has been developed, which eliminates the necessity of using several functions to predict diameter inside bark at different parts of the stem. The variable form taper function is easy to develop and saves computing time. For the data used in this study, it predicted tree profile as a function of height, diameter at breast height, and height from the ground with less bias than many of the taper-estimating systems found in the literature.

T o t a l i n v e n t o l i e s ancl nzethocls of volzime summa7.y zuhich facilitate conzpilation of t w e and log v o l u m e s t o a n y desi7.ed stancla?.d of utilization a?-e neeclecl. I t shoz~ld be possible t o filzd v o l u m e pel. t~e e t o a n y specified s t a n , d a~d o f zitilization exp).essecl a s stzcm p h e i g h t , to11 clib ancl t o e s t i m a t e component zueights in biomass a n d o t h e~ stziclies. T h e fzinctions developed in t h i s pape.1. ancl based zipon clbh, total h e i g h t , ancl section h e i g h t a s a f~aaction of total h e i g h t c a n m e e t nzost o f t h e s e 7.eqzcir.e)nents. Resume Les fonctions de defilement et leur application dans les inventaires forestiers O n ct besoin cl'in.r;entcii).es fo?-estie?.s conzplets e t d e w~hthocles poztv facilitel-la co?npilation cle volzimes ligne7~n: cl'ar.b?~es e n t i e i s o u cle billes COT-~.esponclant ci cles niveazix d'zitilisation c l h t e~n~i n k s . I1 clev?c~it i t~e possible d'kvalzie7~ le volzinze total pa). ~7.b7.e szsivant cles n i v e a u e cl'zitilisation e ep).imks e n t w m e cle hazitezt?. cle sozicke, cle cliam e t r e ?)zininzunz cl'zitilisation, e t / o u cle f action cle l a hazitezcv totale. D e plzcs, la techniqz~e ~l e v 7~a i t p o u v o i~* fozc?-ni7. la p7,opo7.tion clzt volzime total d e la tige conzp7'ise cla~zs les billes d e tozite longzcezt?~ o z~ clianzkt~e, qz~elqzse soit le s y s t & m e cle nzeszivage enzployk (pieel nzeszwe cle planche, pied czcbe 021 pozcls). L a me'thocle dem3ait azissi pozivoi7. &tm zitiliske pozu. clhc~i)*e l'inflzience cles facteznss biologiqzies s z t~ la f o l w e dzc fzit e t s o n clhfilenzent, e t p o z c~ estime?. les co.l?zposantes dzi poicls clans les ktzicles d e bionzasse e t azttres. L e s f o n c t i o n s cle clkf i l e m e n t p~~k s e n t h e s i c i sont baskes szoP le d h p , l a hazitelo. totale e t des f~~a c t i o n s cle lci haliteln3 totale. Elles pezivent ~~e n c o n t 7 -e ) . la plzipa).t cles e r igences knzsmk?*kes.Modern manufacturing methods, coupled with the multiplicity of possible end-uses of a single log in an integrated operation, require that the forester provide detai!ed information regarding the raw material supply. Data for the modern forest inventory must be collected in a manner that permits flexible and comprehensive analyses. Estimates of total volume per acre no longer suffice. Now the volume of material available in certain sizes and qualities must be estimated with high standards of precision and accuracy.A t the 1963 annual meeting of the Canadian In-

A detailed study using seven data sets, two standing tree volume estimating models, and a height–diameter model showed that fit statistics and lack of fit statistics calculated directly from a regression model can be well estimated using simulations of cross validation or double cross validation. These results suggest that cross validation by data splitting and double cross validation provide little, if any, additional information in the process of evaluating regression models.

In many forestry problems additive regression equations are expected because of the nature of the data. This paper discusses the reasons why one does or does not have these additive equations. Mathematical proofs of the theory are given, with a numerical example.