# Table 2 Summary of methods used for crown biomass estimation in this study

Methods Equations for total crown biomass Selection probability Inclusion probability
Simple random sampling
SRS $${\widehat{\tau}}_i=N{B}_i\left(\frac{1}{n}{\displaystyle \sum_{j=1}^n}{y}_{ij}\right)$$ $${\left(\frac{1}{N{B}_i}\right)}^n$$ $$\frac{n}{N{B}_i}$$
SRS-RAT $${\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{j=1}^n}{y}_{ij}}{{\displaystyle {\sum}_{j=1}^n}B{D}_{ij}^2}{\displaystyle \sum_{j=1}^{N{B}_i}}B{D}_{ij}^2$$ $${\left(\frac{1}{N{B}_i}\right)}^n$$ $$\frac{n}{N{B}_i}$$
PPS $${\widehat{\tau}}_i={\displaystyle \sum_{j\in S}}\frac{y_{ij}}{\pi_{ij}^{PPS}}$$ $${\pi}_{ij}=\frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2}$$ $${\pi}_{ij}^{PPS}=1-{\left(1-{\pi}_{ij}\right)}^n$$
Systematic sampling
SYS $${\widehat{\tau}}_i=N{B}_i\left(\frac{1}{n}{\displaystyle \sum_{j=1}^n}{y}_{ij}\right)$$ $$\frac{n}{N{B}_i}$$ $$\frac{n}{N{B}_i}$$
SYS-RAT $${\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{j=1}^n}{y}_{ij}}{{\displaystyle {\sum}_{j=1}^n}B{D}_{ij}^2}{\displaystyle \sum_{j=1}^{N{B}_i}}B{D}_{ij}^2$$ $$\frac{n}{N{B}_i}$$ $$\frac{n}{N{B}_i}$$
Stratified sampling
STR $${\widehat{\tau}}_i={\displaystyle \sum_{h=1}^H}{\displaystyle \sum_{j=1}^{n_h}}\frac{N_{ih}}{n_h}{y}_{ijh}$$ $$\frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2}$$ $$\frac{n_h}{N_{ih}}$$
STR-RAT $${\widehat{\tau}}_i=\frac{{\displaystyle {\sum}_{h=1}^H}{\displaystyle {\sum}_{j=1}^{n_h}}\frac{N_{ih}}{n_h}{y}_{ijh}}{{\displaystyle {\sum}_{h=1}^H}{\displaystyle {\sum}_{j=1}^{n_h}}\frac{N_{ih}}{n_h}B{D}_{ijh}^2}$$ $$\frac{B{D}_{ij}^2}{{\displaystyle {\sum}_{j=1}^{NB}}B{D}_{ij}^2}$$ $$\frac{n_h}{N_{ih}}$$
STR-PPS $${\widehat{\tau}}_{i\left(STR-PPS\right)}={\displaystyle \sum_{h=1}^H}{\displaystyle \sum_{j\in {S}_h}}\frac{y_{ijh}}{\pi_{ijh}^{\left(STR-PPS\right)}}$$ $$\frac{B{D}_{ijh}^2}{{\displaystyle {\sum}_{j=1}^{N{B}_{ih}}}B{D}_{ijh}^2}$$ $${\pi}_{ij}^{STR-PPS}=1-{\left(1-{\pi}_{ij}\right)}^n$$
1. Notation: $${\widehat{\tau}}_i$$ is the estimated total crown biomass for i th tree; y ij is the oven dry weight of j th branch on i th tree; NB i is the number of branches on i th tree; n is number of branches sampled; N ih is the number of branches in h th stratum on i th tree; and n h is the number of branches sampled in h th stratum.