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For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

Oops! Moreover, if $X\in T_\alpha$, then every proper initial segment of $X$ is in the appropriate $T_\beta$. (Thus the set $T_\alpha$ is actually level $\alpha$ of the tree.)

For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

Oops! Also make sure that, if $X\in T_\alpha$, then every proper initial segment of $X$ is in the appropriate $T_\beta$. (Thus the set $T_\alpha$ is actually level $\alpha$ of the tree.)

For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

Oops! Moreover, if $X\in T_\alpha$, they every proper initial segment of $X$ is in the appropriate $T_\beta$. (Thus the set $T_\alpha$ is actually level $\alpha$ of the tree.)

For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

Oops! Moreover, if $X\in T_\alpha$, then every proper initial segment of $X$ is in the appropriate $T_\beta$. (Thus the set $T_\alpha$ is actually level $\alpha$ of the tree.)

For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

For each ordinal $\alpha\lt\omega_1$ construct a countable collection $T_\alpha$ of well-ordered subsets of $\mathbb Q$ of order type $\alpha$ in such a way that, whenever $\beta\lt\alpha\lt\omega_1$, $X\in T_\beta$, and $\sup X\lt r\in\mathbb Q$, there is a set $Y\in T_\alpha$ such that $X$ is an initial segment of $Y$ and $\sup Y\lt r$.

Oops! Moreover, if $X\in T_\alpha$, they every proper initial segment of $X$ is in the appropriate $T_\beta$. (Thus the set $T_\alpha$ is actually level $\alpha$ of the tree.)