*Tetracentron sinense* Oliver, the only species in the Trochodendraceae family (Fu & Bartholomew 2001), mainly distribute in southwestern and central China (Doweld 1998, Fu & Bartholomew 2001, Zhang *et al.* 2019).Its fossils appeared in the Eocene, indicating that this is a relict
species, making it important for the study of ancient flora and
phylogeny in angiosperms. Its importance in ornamental (Zhang *et al*. 2019), medicinal (Wang *et al*. 2006), and furniture products, has resulted in excessive deforestation of *T. sinense*. Thus, the populations are scattered in distribution, and their natural regeneration is very poor (Wang *et al*. 2006, Gan *et al*. 2013).As such, it is currently listed in CITES Appendix III (Convention on
International Trade in Endangered Species of Wild Fauna and Flora, https://cites.org/eng/node/41216), and as a national second-grade protected plant in China (Fu 1992). To date, the conservation and management of the germplasm resources for *T. sinense* have attracted the most attention*.* Over the past 10 years, the community ecology (Tian *et al*. 2018), sporogenesis and gametophyte development (Gan *et al*. 2012), pollination ecology (Gan *et al*. 2013), seed and seedling ecology (Luo *et al*. 2010, Cao *et al*. 2012, Tang *et al*. 2013, Han *et al*. 2015, Li *et al*. 2015), and genetic diversity (Li *et al*. 2016, Han *et al.* 2017, Li *et al*. 2018) of *T. sinense* have been studied*,*with the aim of improving the conservation of its germplasm resources.
Generally, the natural regeneration in plants is a complex process
including some important links of life history, such as the production,
diffusion and germination of seed, seedling settlement and sapling
formation. Among them, the obstacles that occur in any link will lead tothe failure of regeneration (Herrera 1991). To date, there are no relevant reports on the current situation of *T. sinense* natural population, and which link in its life history has the obstacle that causes it poor regeneration is unclear.

Plant population structure is the most basic characteristic of a
population. It can reflect population dynamics and development trends,
in addition to the correlation between a population and its environment,including its actual location within that environment (Chapman & Reiss 2001, Xie *et al*. 2014).Knowledge of plant population structure can illuminate the past and
future trends of a population, and then elucidate the weak links in lifehistory correlated to poor regeneration, which is of relevant for
conservation, management, and utilization of endangered plants (Harper 1977).Life tables and survival curves are important tools for the evaluation
of population dynamics as they show the actual numbers of surviving
individuals, deaths, and survivorship trends for all age classes in the
population. Life tables are also used to explain changes in population
size (Smith & Keyfitz 1947, Skoglund & Verwijst 1989, Armesto *et al.* 1992). Survival curves can elucidate the trends in population changes directly and quickly (Zhang & Zu 1999, Wu *et al*. 2000, Bi *et al*. 2001).The four survival functions (survival rate, cumulative mortality,
mortality density, and hazard rate) can more accurately explain current
population structure and the development parameters of a given
population when combined with life tables (Harcombe 1987, Zhang *et al.* 2008).Time series analysis, an important method in population statistics, is
usually used to forecast population dynamic trends in the future (Xie *et al.* 2014).Therefore, life tables and time series analyses based on the
quantitative characteristics of an endangered plant population, are of
practical importance for the effective conservation and management of
endangered plants (Wu *et al*. 2000, Phama *et al*. 2014).

In this paper, the population structure and derived metrics of four typical *T. sinense*patches in the Leigong Mountain Nature Reserve (LMNR) were studied. Theaims of the study were: (1) analyze the current status of *T. sinense*populations in the LMNR, and to reveal the weak links in life history
correlated to poor regeneration; (2) predict future development trends
of natural *T. sinense* populations in the LMNR; (3) put forward some strategies for the conservation and management of these endangered plants.

### Materials and methods

*Study area.*The LMNR, located in the central part of Qiandongnan, Guizhou Province,China, is made up of steep slopes, which lead into narrow valleys and
small waterfalls. It spans the four counties of Leishan, Taijiang,
Jianhe, and Rongjiang, and is a watershed between the Yangtze River and
the Pearl River (26° 20' 25″-26° 25' 00″ N, 108° 12' 00″-108° 20' 00″ E,2,178 m asl). LMNR is characterized by a typical subtropical humid
climate, with abundant rainfall and less daylight time (Chen *et al．*2012).The mean annual temperature is about 10 °C; July is the hottest month,
with an average temperature of less than 25 °C. The soil is an acidic
mountain yellow soil. The mountain forests are structurally and
floristically heterogeneous (Tang *et al*. 2007, Tang & Ohsawa 2009),with a distinctive vertical distribution of vegetation. The flora is
mainly divided into evergreen broad-leaved forests at lower elevations
(below 1,350 m asl), evergreen and deciduous broad-leaved mixed forests
at middle altitudes (1,350-2,100 m asl), and deciduous broad-leaved
forests at higher altitudes (over 2,100 m asl). The *T. sinense*
populations in the LMNR were mainly scattered throughout evergreen and
deciduous broad-leaved mixed forests. We investigated the distribution
of *T. sinense* throughout the China and found that the distributions of the population of *T. sinense*were patchy. In LMNR, we found that there were four patches which were
the most typical, and they could well represent the distributions of thenatural population of *T. sinense*. Therefore, the four
representative patches were chosen as experimental plots rather than
artificial plots (population 1: 7 ha; population 2: 1.905 ha; population3: 7.4 ha; population 4: 7.02 ha).

*Age structure.* In order to minimize the damage to *T. sinense*,19 individuals with intact growth and different diameter grades were
randomly selected in these four patches and their cores were taken to
determine their ages. Then the relationship between the age and the DBH
of the *T. sinense* were modeled to obtain the fitting curve (Wang *et al*. 1995, Harper 1977, Zhang *et al*. 2007). According to the fitting curve, the age of all individuals in the four populations were calculated (Hett *et al*. 1976). According to the life history characteristics of *T. sinense* and the methods of Brodie *et al*. (1995) and Guedje *et al*. (2003),these populations were grouped into 11 age classes. Pre-reproductive
and juvenile trees were classified as seedlings and saplings (I, 0~20
ages), or juveniles (II, 20~40 ages; III, 40~60 ages; IV, 60~80 ages; V,80~100 ages; VI, 100~120 ages; VII, 120~140 ages). Adult trees were
grouped into four age classes (VIII, 140~160 ages; IX, 160~180 ages; X,
180~200 ages; XI, 200~220 ages). The number of *T. sinense* in each age class was counted, and then the age structure of the populations was analyzed.

*Population dynamics analysis.* The dynamic change in number of individuals between adjacent age classes (*V*_{n
} ) was analyzed according to the method of Chen (1998), using the following formula:

where *S*_{n
} and *S*_{n+1
} are the number of individuals in the *n*^{th
} age class and the next age class, respectively. Max (... ...) represents the maximum value in parentheses; *V*_{n
} ∈ [-1, 1] When *V*_{n
} > 0, this means the number of individuals is increasing in a dynamic relationship between adjacent age classes; when *V*_{n
} < 0, this means there is a decline in the dynamic relationship; when *V*_{n
} = 0, the dynamic relationship is stable.

The quantity dynamic index (*V*_{pi
} ) of the age structure of these populations was obtained by weighting the number of individuals (*S*_{n
} ) of each age class by *V*_{n
} . Because there is no *V*_{n
} for the maximum age class (*K*), the *K* value was excluded (Chen 1998):

Because the external environmental effect on the population age structure is not considered in equation (2), the quantity dynamic index of population age structure (*V*_{pi
} ) should be corrected accordingly, that is:

where min (... ...) means the minimum value of the sequence in parentheses, and *K* is the age class.

*Establishment of a static life table.* Given that *T. sinense*is, a long-life cycles tree species, a static life table was
established. Static life tables capture discrete periods of the dynamic
process of aging where multiple estimate the number of surviving
individuals (x) (Jiang 1992, Zhou *et al.* 1992), based on the following formulas (Silvertown 1982, Jiang 1992).According to the assumptions underlying static life tables, the age
combination is stable, and the proportion of each age class remains
constant (Jiang 1992). Then the data of the four populations in LMNR were corrected by smoothing technique (Proctor 1980, Brodie *et al.* 1995, Molles 2002).

where T: the sum of all the individuals in each group; n: the number of age classes in each group; a_{x}: individuals of each groups in the x age class;
$\overline{{a}_{x}}$
: the means of each group, which were considered to be the
mid-value. Based on the differences between the maximum and minimum
values of number surviving in the two groups and the difference between
the means of the two groups, the number of individuals in each age classwas corrected by one unit of smoothing (Jiang 1992).

where *x*: the age class; *a*_{x
} : number of surviving individuals in age class *x*; *A*_{x
} : number of surviving individuals in age class *x* after smoothing; *A*_{1
} : number of surviving individuals in the I age class after smoothing; *l*_{x
} : standardized number of surviving individuals at the beginning of age class *x* (generally converted to 1,000); *d*_{x
} : standardized number of individuals dying between age classes *x* and *x*+1; *q*_{x
} : mortality rate between age classes *x* and *x*+1; *L*_{x
} : number of surviving individuals between age classes *x* and *x*+1; *T*_{x
} : total number of individuals from age class *x*; *e*_{x
} : life expectancy at age class *x*; *K*_{x
} : killing power; *S*_{x
} : survival rate (Silvertown 1982, Jiang 1992).

*Survival curve.* Taking the *ln*^{lx
} of the number of survivors as the vertical coordinate and each
age class as the horizontal coordinate, the static life table was used
as a basis for the survival curve, which was fitted using SPSS 23.0
software and was modeled by R language (R Core Team 1995). OriginPro 8.0 software was used to draw the mortality rate (*q*_{x
} ) and killing power (*K*_{x
} ) curves (Feng 1983).

According to Hett & Loucks (1976),three kinds of mathematical models (linear equation, exponential
equation, and power function equation) can be used to test of the best
fit of the survival curve on *T. sinense* data (Deevey 1947). These were:

where *x* is age class, *N*_{x
} is the natural logarithm of the standardized number of surviving individuals in age class *x*, *N*_{x
} = ln*l*_{x
} ; *N*_{0} and *b* were directly obtained by fitting the mathematical model.

*Survival analysis.* To better analyze the age structure of the *T. sinense*populations in each patch of LMNR and further clarify the survival
rules for the populations, this study introduced four functions. The
survival functions are functions relating to any age class, which are
more intuitive than the survival curve*,* so survival analysis has agreater practical applied value in the analysis of population life
tables compared to survival curves (Yang *et al*. 1991, Guo 2009). The four functions (population survival rate function *S*_{(i)
} , cumulative mortality function *F*_{(i)
} , mortality density function *f*_{(ti)
} , and hazard rate function *λ*_{(ti)
} were calculated as follows:

where *i* is age class; *S*_{i
} is the survival rate in age class *i*; *S*_{i
} *= S*_{x
} as in equation (2); and *h*_{i
} is the width of the age class.

*Time series analysis.* Time series analysis is often used to predict dynamic changes in population size (Xiao *et al.* 2004, Zhang *et al.* 2017). In this paper, the moving average method was applied to the analysis:

where *n* is the time being predicted (age class period in the study); *M*^{(1)
} _{t
} is the population size in age class *t* after *n* age class periods in the future, and the population size of the current *k* age class of *X*_{k
} . The population quantity dynamics in the next 2, 4, 6, 8, and 10 age class time periods were predicted.

### Results

*Age structure of T. sinense populations*. The fitting results showed that the relationship between the ages and DBH was well fitted by the linear regression (*R*^{2} = 0.826, *P* < 0.01). Therefore, in 95 % prediction range, the family had better fitting observation value. The ages of *T. sinense* in four populations were calculated by the formula (y = 3.15 × x + 21.68) (Figure 1). Then the structures of age were analyzed according to the ages.

Figure 1.

The relationship between DBH and Age

The age structures of the *T. sinense*
populations in the LMNR were all close to the pyramid type, although
their age structures were all incomplete. The maximum number of
individuals in each population was observed in the II or III age class,
there were relatively no seedlings and saplings existed in these
populations except for Population 2 (Figure 2).