Review Global Tree Taper Modelling: A Review of Applications, Methods, Functions, and Their Parameters Serajis Salekin 1,* , Cristian Higuera Catalán 2, Daniel Boczniewicz 2, Darius Phiri 3 , Justin Morgenroth 2 , Dean F. Meason 1 and Euan G. Mason 2 1 Scion-New Zealand Forest Research Institute Ltd., 49 Sala Street, Private Bag 3020, Rotorua 3046, New Zealand; [email protected] 2 New Zealand School of Forestry, University of Canterbury, Christchurch 8140, New Zealand; [email protected] (C.H.C.); [email protected] (D.B.); [email protected] (J.M.); [email protected] (E.G.M.) 3 Department of Plant and Environmental Sciences, School of Natural Resources, Copperbelt University, Kitwe 10101, Zambia; [email protected] * Correspondence: [email protected] Abstract: Taper functions are important tools for forest description, modelling, assessment, and management. A large number of studies have been conducted to develop and improve taper functions; however, few review studies have been dedicated to addressing their development and parameters. This review summarises the development of taper functions by considering their parameterisation, geographic and species-specific limitations, and applications. This study showed that there has been an increase in the number of studies of taper function and contemporary methods have been developed for the establishment of these functions. The reviewed studies also show that taper functions have been developed from simple equations in the early 1900s to complex Citation: Salekin, S.; Catalán, C.H.; functions in modern times. Early taper functions included polynomial, sigmoid, principal component Boczniewicz, D.; Phiri, D.; analysis (PCA), and linear mixed functions, while contemporary machine learning (ML) approaches Morgenroth, J.; Meason, D.F.; Mason, include artificial neural network (ANN) and random forest (RF). Further analysis of the published E.G. Global Tree Taper Modelling: A literature also shows that most of the studies of taper functions have been carried out in Europe Review of Applications, Methods, and the Americas, meaning most taper equations are not specifically applicable to tropical tree Functions, and Their Parameters. species. Developing well-conditioned taper functions requires reducing the variation due to species, Forests 2021, 12, 913. https:// doi.org/10.3390/f12070913 measurement techniques, and climatic conditions, among other factors. The information presented in this study is important for understanding and developing taper functions. Future studies can Academic Editor: Eric Casella focus on developing better taper functions by incorporating emerging remote sensing and geospatial datasets, and using contemporary statistical approaches such as ANN and RF. Received: 20 May 2021 Accepted: 9 July 2021 Keywords: stem shape; taper; growth and yield; forest mensuration; tree structure; forest inventory Published: 13 July 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in 1. Introduction published maps and institutional affil- The concepts of taper functions have long been a part of forestry, and these concepts iations. have been defined and named in various ways in the forest scientific community. The rate of narrowing in stem diameter with increasing height from ground level to the tip of a tree is defined as tree stem taper (e.g., [1,2]). The term ‘taper’ function is often used interchangeably with tree form, where ‘form’ refers to the shape of the tree [1]. Tree stems Copyright: © 2021 by the authors. have multiple inflection points along their length, resulting in multiple geometric shapes [3]. Licensee MDPI, Basel, Switzerland. Consequently, it is difficult to obtain a general mathematical description of the entire tree. This article is an open access article The overall geometric shape can be expressed as a mathematical function of height above distributed under the terms and ground level, total tree height, and diameter at breast height (D) [4]. These mathematical conditions of the Creative Commons functions are generally described as taper functions [1]. Attribution (CC BY) license (https:// Taper functions are essential and play a pivotal role in forest inventory and growth creativecommons.org/licenses/by/ projection, as well as in forest management planning [5]. They can provide a great deal 4.0/). Forests 2021, 12, 913. https://doi.org/10.3390/f12070913 https://www.mdpi.com/journal/forests Forests 2021, 12, x FOR PEER REVIEW 2 of 20 Forests 2021, 12, 913 Taper functions are essential and play a pivotal role in forest inventory and growth2 of 20 projection, as well as in forest management planning [5]. They can provide a great deal of information for decision making at an individual tree level, stand level, and forest level of[6]. information In particular, for these decision functions making provide at an estimates individual of treediameter level, at stand any point level, along and foresta tree levelstem, [total6]. In volume, particular, and these individual functions volumes provide for estimates logs of any of diameterlength at atany any height point from along the a treeground stem, [7–9]. total They volume, are also and used individual to estimate volumes tree forheights logs across of any a length range at of any diameter height meas- from theurements ground [10]. [7– 9]. They are also used to estimate tree heights across a range of diameter measurementsOver the past [10]. century, tree taper functions have been widely studied all over the world.Over Different the past taper century, functions tree taper have functions been developed have been by widely increasing studied the allflexibility over the and world. ap- Differentplicability taper for functionsmany tree have species been developedgrown commercially by increasing (e.g., the flexibility Pinus, Abies, and applicabilityEucalyptus) for[6,11,12]. many This tree speciesis because grown well-developed commercially ta (e.g.,per functionsPinus, Abies, cannot Eucalyptus only give)[6, 11precise,,12]. This unbi- is becauseased estimates well-developed of diameter taper inside functions bark (DIB) cannot or diameter only give outside precise, bark unbiased (DOB), estimatesbut can also of diameterbe easily insideadapted bark for (DIB) a wide or diametervariety of outside species bark and (DOB), generate but accurate can also prediction be easily adapted of tree forstem a widevolume variety [13]. of species and generate accurate prediction of tree stem volume [13]. The search for biologically rational and flexibleflexible taper functions began decadesdecades agoago andand has been stimulated more recently by an increaseincrease inin computationalcomputational resources (Figure 11).). There are a number of notable comparative studiesstudies (e.g., [[6,12,14])6,12,14]) in which different taper functions havehave beenbeen compared compared rigorously rigorously and and the the best best one one chosen chosen for for a given a given scenario; scenario; the bestthe best approaches approaches generally generally vary vary with with region regi andon species.and species. In addition, In addition, basic basic development develop- functions,ment functions, limitations, limitations, and advantages and advantages of the most of the commonly most commonly used taper used functions taper functions are well documentedare well documented [6,15,16], [6,15,16], most recently most inrecently a review in bya review McTague by McTague and Weiskittel and Weiskittel [16]. However, [16]. theirHowever, geographic their geographic distribution, distribution, species-specific specie applicability,s-specific applicability, parameterization, parameterization, and simple yetand useful simple mathematical yet useful mathematical forms have notforms been have comprehensively not been comprehensively reviewed. reviewed. Figure 1. Number of published articles on tapertaper functions and refinementrefinement processprocess ofof thisthis study’sstudy’s literatureliterature search.search. Therefore, thethe mainmain purposes purposes of of this this study study are are as as follows: follows: (a) (a) to to gain gain an an understanding understand- ofing the of geographicthe geographic and and forest forest type type context contex fort the fortaper the taper models models that havethat have been been developed; devel- (b)oped; to describe(b) to describe the evolution the evolution of taper of taper functions functions over over time; time; (c) to (c) quantify to quantify the the accuracy accuracy of taperof taper functions; functions; and and (d) (d) to identifyto identify opportunities opportunities for for new new taper taper function function development. development. To achieveTo achieve these these specific specific goals, goals, this this review review collected collected and and categorised categorised different different taper taper functions func- basedtions based on their on naturetheir nature and development and development with simplewith simple mathematical mathematical forms. forms. It also It recorded also rec- allorded the all available the available parameters parameters for different for different species species and regions. and regions. The information The information presented pre- heresented is usefulhere is foruseful forest for productivity forest productivity projection, projection, forest modelling, forest modelling, and forest and management. forest man- agement. 2. Literature Search and Compilation